{"id":4159,"date":"2022-09-09T10:51:24","date_gmt":"2022-09-09T08:51:24","guid":{"rendered":"https:\/\/erlebnisland-mathematik.de\/?page_id=4159"},"modified":"2023-06-21T11:35:20","modified_gmt":"2023-06-21T09:35:20","slug":"advanced-text-have-the-courage","status":"publish","type":"page","link":"https:\/\/erlebnisland-mathematik.de\/en\/advanced-text-have-the-courage\/","title":{"rendered":"Advanced text Have the courage"},"content":{"rendered":"<div class=\"wpb-content-wrapper\"><p>[vc_row drowwidth=&#8221;sidebar-biest-default sidebar-biest&#8221;][vc_column][vc_column_text]<\/p>\n<h1>Have the courage<\/h1>\n<p> Surely you have already worked with the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Rubik%27s_Cube\" target=\"_blank\" rel=\"noopener\"><em>Rubik&#8217;s Cube<\/em><\/a>, or at least watched others do so. Of course, this cube has a lot to do with mathematics &#8212; but what exactly? The &#8220;<a href=\"https:\/\/erlebnisland-mathematik.de\/en\/?post_type=exponat&amp;p=3516\" rel=\"noopener\">Have the courage<\/a>&#8221; puzzle is also of a similar nature. The commonality is that you can apply certain <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/List_of_transforms\" target=\"_blank\" rel=\"noopener\">transformations<\/a><\/em> one after the other, all of which can be reversed.[\/vc_column_text][vc_single_image image=&#8221;1069&#8243; img_size=&#8221;large&#8221; alignment=&#8221;center&#8221;]Figure 1: The &#8220;Have the courage&#8221; exhibit[\/vc_single_image][vc_column_text]<\/p>\n<h3>And now &#8230; the mathematics:<\/h3>\n<p>[\/vc_column_text][vc_column_text]<\/p>\n<ul>\n<li><a href=\"https:\/\/en.wikipedia.org\/wiki\/Group_theory\" target=\"_blank\" rel=\"noopener\"><em>Group theory<\/em><\/a> is a mathematical discipline that deals precisely with such structures: Given a certain structure <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-a9ca23565ef34461bb8623fb13b713c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"14\" style=\"vertical-align: 0px;\"\/> (for example, the Rubik&#8217;s Cube or the sliding puzzle &#8220;<a href=\"https:\/\/erlebnisland-mathematik.de\/en\/?post_type=exponat&amp;p=3516\" rel=\"noopener\">Have the courage<\/a>&#8220;), which can be <em>transformed<\/em> in a certain way. By this we mean that there are a lot of permissible <em>Operations<\/em> that transform <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-a9ca23565ef34461bb8623fb13b713c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"14\" style=\"vertical-align: 0px;\"\/> into another state, so that two of these operations can be carried out in succession and there is a unique <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Group_theory\" target=\"_blank\" rel=\"noopener\">inverse<\/a><\/em> Operation exists that will restore these <em>undo.<\/em> These operations on the structure <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-a9ca23565ef34461bb8623fb13b713c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"14\" style=\"vertical-align: 0px;\"\/> now generate a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Group_(mathematics)\" target=\"_blank\" rel=\"noopener\"><em>Group<\/em><\/a>. You probably already know many groups from school lessons. Here are a few examples:\n<ul>\n<li>Let us assume that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-fc5f95369df89b5abed0e7d6f6c5d050_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: 0px;\"\/> is the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Real_number\" target=\"_blank\" rel=\"noopener\"><em>real straight line<\/em><\/a> on which we particularly mark out a point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b71cb567b0cc6505ad779a4bc535e3d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#32;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"\/>. We can now apply to this straight line a <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Position_(geometry)#Relative_position\" target=\"_blank\" rel=\"noopener\">displacement<\/a><\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-1b24e405617921274f0d27b011a81e2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\"\/> by a real number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-6b4d17a26029a70c2ed0a9b3c252a028_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\"\/>, i.e. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-1b24e405617921274f0d27b011a81e2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\"\/> transforms the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-82fc2d197e22b7961d3138638812f927_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> into <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-0685e934e3b89bd3b8543afb326d6c60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: -2px;\"\/> and thus in particular <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b71cb567b0cc6505ad779a4bc535e3d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#32;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"\/> into <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2b2876cc004c9e8654eff6acde55883d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#32;&#112;&#43;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"39\" style=\"vertical-align: -4px;\"\/>. Consider that performing the displacement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-1b24e405617921274f0d27b011a81e2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-5ddd35c524d7d4357309cda1bbd02c1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\"\/> in succession is exactly the same as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-c41827a1f4fc2e09fae40f42035dec9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#123;&#114;&#43;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -5px;\"\/>. Therefore, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-1b24e405617921274f0d27b011a81e2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\"\/> can be uniquely reversed by applying <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-ade638dd0830fead065415c734dc44aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#123;&#45;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"25\" style=\"vertical-align: -3px;\"\/>. Thus a set <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-e98ecb133e700bef063e9cd4d27cf7fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#123;&#92;&#116;&#97;&#117;&#95;&#114;&#32;&#92;&#109;&#105;&#100;&#32;&#114;&#92;&#105;&#110;&#32;&#82;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\"\/> of admissible displacements generates a group <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/> operating on the real straight line. Here we can choose the set <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-c03d8b4c07be7629c9005f72332dc867_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#115;&#117;&#98;&#115;&#101;&#116;&#101;&#113;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -3px;\"\/> arbitrarily, so that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b00ab1869efc497aeb72a0ade2fff810_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#92;&#105;&#110;&#32;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"42\" style=\"vertical-align: -1px;\"\/> is also <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-66c40b6f723bfc75a59e65dba1f7a8a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#114;&#92;&#105;&#110;&#32;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"55\" style=\"vertical-align: -1px;\"\/>. For example, if we set <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2acc98851dc1ce22090792ee5735aa65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#123;&#45;&#49;&#44;&#49;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -5px;\"\/>, we get exactly the <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Integer\" target=\"_blank\" rel=\"noopener\">integer<\/a><\/em> shifts for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/>: Each shift by an <em>integer<\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b0856924b7dbf0e0e210f03b8c38c79a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#90;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\"\/> can be understood as a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-3e251b15ab0e15357b84ad4c49091215_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#122;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"15\" style=\"vertical-align: -5px;\"\/>-fold execution of the shift <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-3bf3652042511b61666d4716a8c975c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#115;&#103;&#110;&#125;&#40;&#122;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"44\" style=\"vertical-align: -8px;\"\/>. However, we can also choose <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-a2c1d2fb2fba2d9652e0ba702eba4f66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#91;&#45;&#49;&#44;&#49;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" style=\"vertical-align: -5px;\"\/>. Then we actually obtain for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/> all displacements by arbitrary <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Real_number\" target=\"_blank\" rel=\"noopener\">real numbers<\/a><\/em>: Any displacement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-5ddd35c524d7d4357309cda1bbd02c1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\"\/> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-fc275ebd4fb9eaa9267ba6ec13efefee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\"\/> arbitrary, can be understood as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-514875502eaca39f18b3bb51718fae81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#99;&#101;&#105;&#108;&#124;&#115;&#124;&#92;&#114;&#99;&#101;&#105;&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"28\" style=\"vertical-align: -5px;\"\/>-fold of the displacement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-1b24e405617921274f0d27b011a81e2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\"\/> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-16e7b2f6a15d9ebcd940a3847396ed40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#115;&#47;&#92;&#108;&#99;&#101;&#105;&#108;&#124;&#115;&#124;&#92;&#114;&#99;&#101;&#105;&#108;&#92;&#105;&#110;&#91;&#45;&#49;&#44;&#49;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" style=\"vertical-align: -5px;\"\/>.<\/li>\n<li>Similarly, you can also create a group by <a href=\"https:\/\/en.wikipedia.org\/wiki\/Rotation_(mathematics)\" target=\"_blank\" rel=\"noopener\"><em>turning<\/em><\/a>. For this we take as structure <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-a9ca23565ef34461bb8623fb13b713c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"14\" style=\"vertical-align: 0px;\"\/> the <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Unit_circle\" target=\"_blank\" rel=\"noopener\">unit circle<\/a><\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aede64402ad4f5f55845e32f579dec0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#61;&#92;&#123;&#40;&#120;&#44;&#121;&#41;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#82;&#94;&#50;&#92;&#44;&#124;&#92;&#44;&#120;&#94;&#50;&#43;&#121;&#94;&#50;&#61;&#49;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"238\" style=\"vertical-align: -5px;\"\/> together with the excellent point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b71cb567b0cc6505ad779a4bc535e3d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#32;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"\/>. Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-832f2d6cc45ae441193a9ea60148ddb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#95;&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"\/> be the rotation by the angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-89002a65c61e6d4d9e7493669fb5a416_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: -1px;\"\/> (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Radian\" target=\"_blank\" rel=\"noopener\"><em>in radians<\/em><\/a>). As above, one convinces oneself that the succession of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-832f2d6cc45ae441193a9ea60148ddb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#95;&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-902a90b56fc496501b261ff418d16ca6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#95;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -6px;\"\/> yields exactly <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-d74436173759a24c417f3ca5b897a772_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#95;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#43;&#92;&#98;&#101;&#116;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"37\" style=\"vertical-align: -6px;\"\/> (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-49421c344d506037913951244a6ed420_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"65\" style=\"vertical-align: -4px;\"\/>). Again, we can consider the group <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/> generated by the rotations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-9ae2cb3da341fe8c6697d8e884e48187_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#123;&#92;&#114;&#104;&#111;&#95;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#109;&#105;&#100;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#92;&#105;&#110;&#32;&#82;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -5px;\"\/> (for a set <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-c03d8b4c07be7629c9005f72332dc867_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#115;&#117;&#98;&#115;&#101;&#116;&#101;&#113;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -3px;\"\/> of <em>admissible<\/em> angles such that with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-666c63826987de2a767c102000c00799_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#92;&#105;&#110;&#32;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"46\" style=\"vertical-align: -1px;\"\/> also <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-0d99df6c82675dacd197c11f82dc9219_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#97;&#108;&#112;&#104;&#97;&#92;&#105;&#110;&#32;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"59\" style=\"vertical-align: -1px;\"\/>). For example, if we take <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b30b9e8cb1bd82dcd2c9f0b257af9c80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#123;&#45;&#50;&#32;&#92;&#112;&#105;&#47;&#110;&#44;&#50;&#92;&#112;&#105;&#47;&#110;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -5px;\"\/>, we get for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/> exactly those rotations by a multiple of the angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-0ede9dd212785eb1ed178596c7bb5a27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#105;&#47;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"\/>. These are exactly the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Orientation_(vector_space)#Definition\" target=\"_blank\" rel=\"noopener\"><em>orientation-preserving<\/em><\/a> self-images of the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Regular_polygon\" target=\"_blank\" rel=\"noopener\"><em>regular <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8ad02d5f0d297c7adacb6b8c4f9fee74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/>-corner.<\/em><\/a> But we could also choose <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-c47d449804809be6dcececbb2768cb91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#40;&#45;&#92;&#112;&#105;&#44;&#92;&#112;&#105;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -5px;\"\/> and in this way get for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/> all rotations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-832f2d6cc45ae441193a9ea60148ddb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#95;&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"\/> around the origin.<\/li>\n<li>If we take as structure <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-a9ca23565ef34461bb8623fb13b713c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"14\" style=\"vertical-align: 0px;\"\/> a deck of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8ad02d5f0d297c7adacb6b8c4f9fee74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/> cards laid out face up on a table, we can consider as operations certain permissible <a href=\"https:\/\/en.wikipedia.org\/wiki\/Permutation\" target=\"_blank\" rel=\"noopener\"><em>rearrangements<\/em><\/a> of these cards. If, for example, one chooses as permissible rearrangements the swaps of two adjacent cards each, these create a group: in fact, each of these rearrangements can be undone by simply applying it again. However, one also considers that any rearrangement of the cards can be combined from the pairwise swaps of two adjacent cards. This group of all permutations is called the fully <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Symmetric_group\" target=\"_blank\" rel=\"noopener\">symmetric group<\/a><\/em> of degree <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8ad02d5f0d297c7adacb6b8c4f9fee74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/>.<\/li>\n<\/ul>\n<p>There are many more such examples. Maybe you try to find some yourself. The exhibit &#8220;<a href=\"https:\/\/erlebnisland-mathematik.de\/en\/?post_type=exponat&amp;p=3516\" data-rich-text-format-boundary=\"true\">Have the courage<\/a>&#8221; also hides a special group. This <a href=\"https:\/\/en.wikipedia.org\/wiki\/Permutation\" target=\"_blank\" rel=\"noopener\"><em>permutes<\/em><\/a> and turns the nine tiles with the letters. In the following, we want to understand this group &#8212; and thus the exhibit &#8212; in more detail.<\/li>\n<\/ul>\n<p>[\/vc_column_text][vc_column_text]<\/p>\n<h3>About the exhibit &#8220;<a href=\"https:\/\/erlebnisland-mathematik.de\/en\/?post_type=exponat&amp;p=3516\">Have the courage<\/a>&#8220;<\/h3>\n<p>[\/vc_column_text][vc_column_text]But which group exactly is hidden behind the exhibit? Let&#8217;s understand that better now. Given nine similar tiles arranged in a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-6397976089e6b81a9c4d73d986f8b8d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\"\/> square, each of which we can <a href=\"https:\/\/en.wikipedia.org\/wiki\/Cyclic_permutation\" target=\"_blank\" rel=\"noopener\"><em>rotate<\/em><\/a> four, forming a small <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2b4cfc2ebe6fd792c107b410e45967e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\"\/> square at one corner. However, the individual parts are also tilted by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-88fd5a1e3a9181f9da1d254c041c08d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#48;&#94;&#92;&#99;&#105;&#114;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\"\/> each (see figure 1 below).[\/vc_column_text][vc_single_image][\/vc_single_image][vc_column_text]All other operations are now composed &#8212; as in the examples above &#8212; of these four rotations, i.e. our <em>group<\/em> of rearrangements is <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Dihedral_group_of_order_6#Summary_of_group_operations\" target=\"_blank\" rel=\"noopener\">generated<\/a><\/em> by the four rotations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/>. If one performs each of these rotations four times in succession, one returns to the initial state, i.e. the <a href=\"https:\/\/de.wikipedia.org\/wiki\/S3_(Gruppe)#Erzeuger_und_Relationen\" target=\"_blank\" rel=\"noopener\"><em>relations<\/em><\/a> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-4faed056027f28e148b4b3f03c212c8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#52;&#61;&#121;&#94;&#52;&#61;&#122;&#94;&#52;&#61;&#119;&#94;&#52;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"180\" style=\"vertical-align: -4px;\"\/> (the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Identity_function\" target=\"_blank\" rel=\"noopener\"><em>identical figure<\/em><\/a>) apply. But can any conceivable arrangement of the nine parts also be achieved by performing the four turns one after the other? That is, can any rearrangement and subsequent twisting of the nine parts be combined from the rotations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/>?<\/p>\n<p>This is the question we want to explore in the following. (Mathematically expressed: <em>Which subgroup of the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Wreath_product\" target=\"_blank\" rel=\"noopener\">wreath product<\/a> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-fd20c3410fdd1db93709ab234e9f565a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#95;&#52;&#92;&#119;&#114;&#32;&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -3px;\"\/> of the cyclic group with four elements, corresponding to the possible rotations of a tile, and the symmetric group on nine digits, corresponding to the possible <a href=\"https:\/\/en.wikipedia.org\/wiki\/Permutation\" target=\"_blank\" rel=\"noopener\">permutations<\/a> of the tiles, produce the rotations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/>?)<\/em><\/p>\n<p>To do this, we number the tiles from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-07de1e9aebc049111b4109ae367f3ba6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"7\" style=\"vertical-align: 0px;\"\/> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-ecec689b819efb8fe33a9af8d6b91805_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/> and first identify them with each other according to Figure 2 below:[\/vc_column_text][vc_single_image][\/vc_single_image][vc_column_text]So what does this identification mean and why is our chosen identification so well suited to the study of the exhibit? Now, the identification allows us to add to each <em>permutation<\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-6cd4037c095747ce6dd8a25325124a87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#103;&#109;&#97;&#92;&#105;&#110;&#32;&#83;&#95;&#57;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#83;&#121;&#109;&#125;&#40;&#92;&#123;&#49;&#44;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#57;&#92;&#125;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"188\" style=\"vertical-align: -5px;\"\/> to associate a rearrangement of the exhibit by transferring the platelet <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-6bcb0de135c476d4cc7796bd7ee97d13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"5\" style=\"vertical-align: 0px;\"\/> into the platelet <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8fde4571e00c3059b7cafff9db77d11e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#103;&#109;&#97;&#40;&#105;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"30\" style=\"vertical-align: -5px;\"\/> &#8212; according to the chosen identification of the platelets in Figure 2. In this way we thus obtain an\u00a0<a href=\"https:\/\/en.wikipedia.org\/wiki\/Injective_function\" target=\"_blank\" rel=\"noopener\"><em>embedding<\/em><\/a> of the symmetric group <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-d7376707492eccc0bbd2a55feb1eb7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -3px;\"\/> to nine digits in the group of all rearrangements of our exhibit. In addition, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-4e1fc5a813d7e0fe9573ea4a1a0bf5af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"7\" style=\"vertical-align: 0px;\"\/> be the rearrangement that twists only the middle tile (number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-ba00d47237913306b025fcd1d5bdd43f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/>) by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-159dfc4bc61c7fbd640d439e21e370c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;&#94;&#92;&#99;&#105;&#114;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"33\" style=\"vertical-align: 0px;\"\/> (so that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-86a091f51f3e7cf8b10ecee149b6e017_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#94;&#50;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: 0px;\"\/> is the identical rearrangement). Our chosen identification is so convenient because the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Dihedral_group_of_order_6#Summary_of_group_operations\" target=\"_blank\" rel=\"noopener\"><em>generators<\/em><\/a> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/> can now be expressed very nicely using the chosen copy of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-d7376707492eccc0bbd2a55feb1eb7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -3px;\"\/> and the rotation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-4e1fc5a813d7e0fe9573ea4a1a0bf5af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"7\" style=\"vertical-align: 0px;\"\/>:<\/p>\n<p>The rotation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-82fc2d197e22b7961d3138638812f927_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Rotation_(mathematics)\" target=\"_blank\" rel=\"noopener\">rotates<\/a><\/em> the platelets <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-c90a50930c9d6676aa358c91f2a2e50b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#50;&#44;&#53;&#44;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: -3px;\"\/> cyclically and corresponds exactly to the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-66a2fa87506de0b52df08512caab6bda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/>-cycle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-5ad4949208f729ee8773f557044afecd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#49;&#44;&#50;&#44;&#52;&#44;&#53;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"\/> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-d7376707492eccc0bbd2a55feb1eb7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -3px;\"\/> (because it is compatible with the chosen identification of the platelets; here we use the <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Permutation#Cycle_notation\" target=\"_blank\" rel=\"noopener\">cycle notation<\/a><\/em> for <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Permutation\" target=\"_blank\" rel=\"noopener\">permutations<\/a><\/em>). Similarly, we also obtain that exactly <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-597240fbe4c620ebc2360210919e7b11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#40;&#53;&#44;&#54;&#44;&#57;&#44;&#56;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"\/> holds. With the other rotations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-7d66fb62a1b86120e9f14f3ac7cbaeaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-82fc2d197e22b7961d3138638812f927_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> we have to be be more careful. For example, if we compare the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-66a2fa87506de0b52df08512caab6bda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/> cycle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-548898e581e8cec9850f0c5be3d17744_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#50;&#44;&#51;&#44;&#54;&#44;&#53;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"\/> with the rotation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-7d66fb62a1b86120e9f14f3ac7cbaeaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"\/>, we see that the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-ed7ac17bcea640a1077dd6b2716b1564_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b64485cf3f6c7fa07b992e8ad9d4497f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/> platelets are mapped the same under both rearrangements, but the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-ba00d47237913306b025fcd1d5bdd43f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8da23762a6f92d3265493fa3767cd0e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/> platelets are rotated to the same place, but end up opposite in orientation by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-159dfc4bc61c7fbd640d439e21e370c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;&#94;&#92;&#99;&#105;&#114;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"33\" style=\"vertical-align: 0px;\"\/> under each of the two rearrangements. Therefore, consider that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-5d8bf208ad85d46027eb99a8bfb1d390_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#116;&#94;&#123;&#45;&#49;&#125;&#40;&#50;&#44;&#51;&#44;&#54;&#44;&#53;&#41;&#116;&#61;&#116;&#40;&#50;&#44;&#51;&#44;&#54;&#44;&#53;&#41;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"253\" style=\"vertical-align: -5px;\"\/>. In the same way, one also obtains that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-cf87156c28cbde3c99676d7f37e05653_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#116;&#40;&#52;&#44;&#53;&#44;&#56;&#44;&#55;&#41;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -5px;\"\/> holds. All this is again summarised in the following figure 3:[\/vc_column_text][vc_single_image][\/vc_single_image][vc_column_text]From the considerations just given, it follows that the group of rearrangements <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/>, which is generated by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/>, is a subgroup of the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Wreath_product\" target=\"_blank\" rel=\"noopener\"><em>wrath product<\/em><\/a> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-e67844782ad2763dbf6385f3ba0a9b3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#116;&#92;&#114;&#97;&#110;&#103;&#108;&#101;&#92;&#119;&#114;&#32;&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -5px;\"\/>. Here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-d7376707492eccc0bbd2a55feb1eb7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -3px;\"\/> is the &#8220;<em>copy<\/em>&#8221; chosen according to the identification in Figure 2 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-0e367df87c09db3d0b5e9b53217f30bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#116;&#92;&#114;&#97;&#110;&#103;&#108;&#101;&#92;&#99;&#111;&#110;&#103;&#32;&#67;&#95;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -5px;\"\/> is the <em>group with two elements generated<\/em> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-4e1fc5a813d7e0fe9573ea4a1a0bf5af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"7\" style=\"vertical-align: 0px;\"\/>.<\/p>\n<p>But which subgroup do we get now exactly? To this end, we note that any rearrangement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8563ce02dc8080c4916841d6a665e6fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#105;&#110;&#32;&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\"\/> that can be generated from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/> can be uniquely written as a product <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-ae8d006157021a49f941dd4581229ef8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#61;&#92;&#115;&#105;&#103;&#109;&#97;&#46;&#114;&#92;&#105;&#110;&#32;&#83;&#95;&#57;&#92;&#108;&#116;&#105;&#109;&#101;&#115;&#32;&#67;&#95;&#50;&#94;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"139\" style=\"vertical-align: -5px;\"\/> (as an element of the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Semidirect_product\" target=\"_blank\" rel=\"noopener\"><em>semidirect product<\/em><\/a>). Here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-f1b369ec1d4cd9e97c00ff1b2c3b8ff0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#103;&#109;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> transfers each tile <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-6bcb0de135c476d4cc7796bd7ee97d13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"5\" style=\"vertical-align: 0px;\"\/> to the desired position <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8fde4571e00c3059b7cafff9db77d11e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#103;&#109;&#97;&#40;&#105;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"30\" style=\"vertical-align: -5px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-94672b7526308efaca1985a5ecdae780_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"7\" style=\"vertical-align: 0px;\"\/> then rotates each individual tile so that it gets the desired orientation (i.e. no longer changes the position of the tiles). We call <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-f1b369ec1d4cd9e97c00ff1b2c3b8ff0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#103;&#109;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> the <em>permutation part<\/em> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-6fbb01a405b8afa99c49c4229a6d60be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -4px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-94672b7526308efaca1985a5ecdae780_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"7\" style=\"vertical-align: 0px;\"\/> the <em>rotation part.<\/em><\/p>\n<p>First, we want to understand the <em>permutation<\/em> part of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/>, i.e. the possible rearrangements we can generate from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/> if we neglect the rotations of the individual platelets (i.e. we want to understand the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Quotient_group\" target=\"_blank\" rel=\"noopener\"><em>quotient<\/em><\/a> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-6e508ceaf4dcf20d6237206dc16f5309_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#71;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"14\" style=\"vertical-align: 0px;\"\/> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/> along the natural mapping <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-0b288c08fee5189a0c4976a47b7cb613_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#116;&#92;&#114;&#97;&#110;&#103;&#108;&#101;&#92;&#119;&#114;&#32;&#83;&#95;&#57;&#92;&#116;&#111;&#32;&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -5px;\"\/>). We claim that we can combine all permutations of the nine tiles of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/> that are possible at all (i.e. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-0777594d93339779da5c10c93e89ef20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#71;&#125;&#61;&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\"\/>):<\/p>\n<p>To do this, we calculate the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-a354eaee36dc06d039ab854ebb8aeeb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#91;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#120;&#125;&#44;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#121;&#125;&#93;&#61;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#120;&#121;&#120;&#94;&#123;&#45;&#49;&#125;&#121;&#94;&#123;&#45;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"135\" style=\"vertical-align: -5px;\"\/> (the so-called <a href=\"https:\/\/en.wikipedia.org\/wiki\/Commutator\" target=\"_blank\" rel=\"noopener\"><em>commutator<\/em><\/a> of the permutations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-967d8c66837da5c5b39f52ea3711121b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"10\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b77f44cafc27d41db875c19e8dea4282_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"11\" style=\"vertical-align: -4px;\"\/>): <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-021714135f7e259675d9dd4d362f0584_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#120;&#121;&#125;&#61;&#40;&#49;&#44;&#51;&#44;&#54;&#44;&#53;&#44;&#52;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -5px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-c54d2d1f48e2de23068213e5e19742b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#120;&#94;&#123;&#45;&#49;&#125;&#121;&#94;&#123;&#45;&#49;&#125;&#125;&#61;&#40;&#49;&#44;&#52;&#44;&#54;&#44;&#51;&#44;&#50;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -5px;\"\/>, and thus <\/p>\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 22px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-93d78d1fdab19af20c471df043dd328e_l3.png\" height=\"22\" width=\"383\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#120;&#121;&#120;&#94;&#123;&#45;&#49;&#125;&#121;&#94;&#123;&#45;&#49;&#125;&#125;&#61;&#40;&#49;&#44;&#51;&#44;&#54;&#44;&#53;&#44;&#52;&#41;&#40;&#49;&#44;&#52;&#44;&#54;&#44;&#51;&#44;&#50;&#41;&#61;&#40;&#49;&#44;&#50;&#41;&#40;&#53;&#44;&#54;&#41;&#46;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>Through <a href=\"https:\/\/en.wikipedia.org\/wiki\/Conjugacy_class\" target=\"_blank\" rel=\"noopener\"><em>conjugation<\/em><\/a> (i.e. shifting the four digits <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-bd43965fda75351796bcf0fb204ae354_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#50;&#44;&#53;&#44;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: -3px;\"\/> by means of precomposition with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2c74668ea689942c5a494e78557f6872_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#103;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"28\" style=\"vertical-align: -4px;\"\/> and composition with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-da3551d4fac694c8c04372eb55700207_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"11\" style=\"vertical-align: -4px;\"\/> for a suitable rearrangement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8563ce02dc8080c4916841d6a665e6fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#105;&#110;&#32;&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\"\/>), any permutation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-1b61b8a7235ba1981fd9aedb79961af5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#97;&#44;&#98;&#41;&#40;&#99;&#44;&#100;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -5px;\"\/> (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-a19d25c184243509a065d692b592b003_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;&#44;&#100;&#92;&#105;&#110;&#92;&#123;&#49;&#44;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#57;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"151\" style=\"vertical-align: -5px;\"\/> pairwise different) can now be obtained (i.e. of the same <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Permutation#Cycle_type\" target=\"_blank\" rel=\"noopener\">type of cycle<\/a><\/em>). But this means that the entire <a href=\"https:\/\/en.wikipedia.org\/wiki\/Conjugacy_class\" target=\"_blank\" rel=\"noopener\"><em>conjugacy class<\/em><\/a> of the element <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-467079ba070d6c96c2073d265e68d986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#49;&#44;&#50;&#41;&#40;&#53;&#44;&#54;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"\/> under the symmetrical group on nine digits <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-d7376707492eccc0bbd2a55feb1eb7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -3px;\"\/> is contained in the group generated by the four base rotations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/>. Thus, this must contain at least the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Alternating_group\" target=\"_blank\" rel=\"noopener\"><em>alternating group<\/em><\/a> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b22e091c59d71eec3ccd8a0744de1836_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -3px;\"\/> generated by it (because this is <a href=\"https:\/\/en.wikipedia.org\/wiki\/Simple_group\" target=\"_blank\" rel=\"noopener\"><em>simple<\/em><\/a>). However, since each of the rotations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/> is a cycle of even length, i.e. yields an <em><a href=\"https:\/\/de.wikipedia.org\/wiki\/Vorzeichen_(Permutation)\" target=\"_blank\" rel=\"noopener\">odd permutation<\/a><\/em>, the generated group must even be all <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-d7376707492eccc0bbd2a55feb1eb7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -3px;\"\/>. Thus, all conceivable permutations of the tiles from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/> can be combined. So we only need to understand the possible <em>rotational proportions<\/em> of elements <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8563ce02dc8080c4916841d6a665e6fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#105;&#110;&#32;&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\"\/>.<\/p>\n<p>We will now devote ourselves to this task: First, we observe that for all generators <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/>, the rotation fractions of the individual plates add up to zero (i.e. only an even number of plates are turned over by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-159dfc4bc61c7fbd640d439e21e370c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;&#94;&#92;&#99;&#105;&#114;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"33\" style=\"vertical-align: 0px;\"\/>). In fact, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-4e4f70906d0cf9505e0296a083f0e1b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#40;&#49;&#44;&#50;&#44;&#52;&#44;&#51;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" style=\"vertical-align: -5px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-597240fbe4c620ebc2360210919e7b11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#40;&#53;&#44;&#54;&#44;&#57;&#44;&#56;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"\/> &#8212; so these elements have no rotational components at all (i.e., in particular, zero <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-d8d89471eee64175b523f99560768183_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"5\" width=\"13\" style=\"vertical-align: 2px;\"\/> just many platelets are re-rotated by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-159dfc4bc61c7fbd640d439e21e370c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;&#94;&#92;&#99;&#105;&#114;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"33\" style=\"vertical-align: 0px;\"\/>). For the elements <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-3d6078c2ae4ed43da3b460cf3ad04e38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#44;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: -4px;\"\/> we get: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-f804e9e5a1635e552642dc91b64ddd50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#116;&#40;&#50;&#44;&#51;&#44;&#54;&#44;&#53;&#41;&#116;&#61;&#40;&#50;&#44;&#51;&#44;&#54;&#44;&#53;&#41;&#116;&#94;&#123;&#40;&#50;&#44;&#51;&#44;&#54;&#44;&#53;&#41;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"285\" style=\"vertical-align: -5px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-4df5c160d2edb8cc68e0fed9ba90a184_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#116;&#40;&#52;&#44;&#53;&#44;&#56;&#44;&#55;&#41;&#116;&#61;&#40;&#52;&#44;&#53;&#44;&#56;&#44;&#55;&#41;&#116;&#94;&#123;&#40;&#52;&#44;&#53;&#44;&#56;&#44;&#55;&#41;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"284\" style=\"vertical-align: -5px;\"\/> (here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-c1677ed0cdb4e754e7ae6b5991cbaea6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#94;&#98;&#61;&#97;&#94;&#123;&#45;&#49;&#125;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"76\" style=\"vertical-align: 0px;\"\/> means the element <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-522dc16829b6bc97d06ce10c3e5dabb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\"\/> <a href=\"https:\/\/en.wikipedia.org\/wiki\/Conjugacy_class\" target=\"_blank\" rel=\"noopener\"><em>conjugated<\/em><\/a> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-257a8ff50082f387c96ed391c555a28d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/>). Thus, at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-7d66fb62a1b86120e9f14f3ac7cbaeaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"\/> the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-ed7ac17bcea640a1077dd6b2716b1564_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-ba00d47237913306b025fcd1d5bdd43f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/> tiles and at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8399287e1380c4dedb435ab147bf7a92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"\/> the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-ba00d47237913306b025fcd1d5bdd43f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b8dce632f927fa35bf53eb13dc7109b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/> tiles are re-rotated by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-159dfc4bc61c7fbd640d439e21e370c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;&#94;&#92;&#99;&#105;&#114;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"33\" style=\"vertical-align: 0px;\"\/> (i.e. an even number).<\/p>\n<p>Consider further that the condition that all rotations of the individual tiles sum to the identical rotation defines a subgroup <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-da3d7367d698288aefda3e09ffe11faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\"\/> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-a8756abe76f256eae773cfe9dc87d7b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#95;&#50;&#94;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"18\" style=\"vertical-align: -5px;\"\/> invariant under permutation of the &#8220;<em>coordinates<\/em>&#8221; (i.e., the tiles <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-07de1e9aebc049111b4109ae367f3ba6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"7\" style=\"vertical-align: 0px;\"\/> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-ecec689b819efb8fe33a9af8d6b91805_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/>) (i.e., <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-e1ff3ef2d7ece32a2692e435def00801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#40;&#117;&#95;&#49;&#44;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#117;&#95;&#57;&#41;&#92;&#105;&#110;&#32;&#85;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"\/> exactly when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-38a1577157ee54c59bd9466449857284_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#49;&#92;&#99;&#100;&#111;&#116;&#115;&#32;&#117;&#95;&#57;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -3px;\"\/>). With this one calculates that then all rotational components of elements from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/> must lie in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-da3d7367d698288aefda3e09ffe11faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\"\/> (this follows from the invariance of the condition under permutation of the nine tiles and from the fact that the rotational components of the generators <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/> lie in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-da3d7367d698288aefda3e09ffe11faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\"\/>).<\/p>\n<p>We now even claim that every rotation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-c81bf366b080a46f3d4267cba7419741_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#92;&#105;&#110;&#32;&#85;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -1px;\"\/> lies in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/>, i.e. can be combined from the generators <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/>. Thus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-75b2d2c6d766503d1fff84410071a689_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#61;&#85;&#92;&#99;&#100;&#111;&#116;&#32;&#83;&#95;&#57;&#61;&#85;&#92;&#114;&#116;&#105;&#109;&#101;&#115;&#32;&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"155\" style=\"vertical-align: -3px;\"\/> must then hold and we can check exactly whether a given rearrangement of the exhibit can be combined from the rotations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/> by first &#8220;<em>subtracting<\/em>&#8221; the permutation part and then checking whether the remaining rotation part satisfies the above condition (i.e. lies in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-da3d7367d698288aefda3e09ffe11faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\"\/>).<\/p>\n<p>So it remains to show that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-da3d7367d698288aefda3e09ffe11faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\"\/> is a subgroup of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/>. To do this, we calculate the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-a9f846265478a463944f061aa7e96589_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#91;&#120;&#44;&#121;&#93;&#41;&#94;&#50;&#61;&#40;&#120;&#121;&#120;&#94;&#123;&#45;&#49;&#125;&#121;&#94;&#123;&#45;&#49;&#125;&#41;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"176\" style=\"vertical-align: -5px;\"\/> (i.e. the square of the commutator of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-82fc2d197e22b7961d3138638812f927_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-7d66fb62a1b86120e9f14f3ac7cbaeaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"\/>) in Figure 4 below:[\/vc_column_text][vc_single_image][\/vc_single_image][vc_column_text]The calculation shows that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-532e58b54cd009f77c9fe5c414b8a2bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#61;&#40;&#91;&#120;&#44;&#121;&#93;&#41;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"93\" style=\"vertical-align: -5px;\"\/> leaves all the tiles in place and flips exactly the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b00736279b18fc0c6cf09b9a26807742_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"24\" style=\"vertical-align: -3px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-c345ccfd80199225d1a1a8f0ee8e5584_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#44;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: -3px;\"\/> tiles by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-159dfc4bc61c7fbd640d439e21e370c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;&#94;&#92;&#99;&#105;&#114;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"33\" style=\"vertical-align: 0px;\"\/>. So <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-532e58b54cd009f77c9fe5c414b8a2bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#61;&#40;&#91;&#120;&#44;&#121;&#93;&#41;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"93\" style=\"vertical-align: -5px;\"\/> lies in the subgroup <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-da3d7367d698288aefda3e09ffe11faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\"\/> and is not the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Identity_element\" target=\"_blank\" rel=\"noopener\"><em>neutral element<\/em><\/a> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-786389d8f492120634a113fe8d30eb97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/>, which does not change any platelet. However, from <a href=\"https:\/\/en.wikipedia.org\/wiki\/Group_representation\" target=\"_blank\" rel=\"noopener\"><em>representation theory<\/em><\/a> we know that the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Module_(mathematics)\" target=\"_blank\" rel=\"noopener\"><em><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-d7376707492eccc0bbd2a55feb1eb7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -3px;\"\/>-module<\/em><\/a> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-da3d7367d698288aefda3e09ffe11faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\"\/> is irreducible, i.e. the subgroup generated invariant by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-6fbb01a405b8afa99c49c4229a6d60be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -4px;\"\/> under <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-d7376707492eccc0bbd2a55feb1eb7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -3px;\"\/> must be all <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-da3d7367d698288aefda3e09ffe11faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\"\/>. But with this we have shown that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-543743c88b8e016e99d29dacf802a7d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#103;&#94;&#123;&#83;&#95;&#57;&#125;&#92;&#114;&#97;&#110;&#103;&#108;&#101;&#61;&#85;&#92;&#115;&#117;&#98;&#115;&#101;&#116;&#101;&#113;&#32;&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"112\" style=\"vertical-align: -5px;\"\/> holds (here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-cd6219c4430081b7f7fcbd702b9a7c3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#94;&#123;&#83;&#95;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -4px;\"\/> is the set of conjugates of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-6fbb01a405b8afa99c49c4229a6d60be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -4px;\"\/> lower elements of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-d7376707492eccc0bbd2a55feb1eb7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -3px;\"\/>), which was to be shown. We have thus fully understood the group <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/>. It is therefore easy to check whether we can return a given arrangement of the exhibit to its original state.<\/p>\n<p>The number of admissible (i.e. combinable out of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/>) arrangements is thus equal to the size of the group <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/>, i.e. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-cccb32b7bfc8954dc82e43cee8381cf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#71;&#124;&#61;&#124;&#85;&#124;&#124;&#83;&#95;&#57;&#124;&#61;&#50;&#94;&#56;&#92;&#99;&#100;&#111;&#116;&#32;&#57;&#33;&#61;&#57;&#50;&#56;&#57;&#55;&#50;&#56;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"259\" style=\"vertical-align: -5px;\"\/> (here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-51c975b3897720eb33f999fce4dbe422_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#33;&#61;&#110;&#92;&#99;&#100;&#111;&#116;&#40;&#110;&#45;&#49;&#41;&#92;&#99;&#100;&#111;&#116;&#115;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -5px;\"\/> is the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Factorial\" target=\"_blank\" rel=\"noopener\"><em>factorial function<\/em><\/a>). If we are very precise, we note that the letters <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-7885983a232453d9a4cb3ddae11da2bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#72;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-803ad811dcdaf6a579e0c019ad8779f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#78;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\"\/> are <a href=\"https:\/\/en.wikipedia.org\/wiki\/Point_reflection\" target=\"_blank\" rel=\"noopener\"><em>point-symmetrical<\/em><\/a>. So whether or not these two perform a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-159dfc4bc61c7fbd640d439e21e370c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;&#94;&#92;&#99;&#105;&#114;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"33\" style=\"vertical-align: 0px;\"\/> rotation doesn&#8217;t matter. Therefore, there would then only be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b3d4d99bec01ede4eb3c27b36d4029db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#71;&#124;&#47;&#50;&#61;&#52;&#54;&#52;&#52;&#56;&#54;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"135\" style=\"vertical-align: -5px;\"\/> possible arrangements (because, if you &#8220;<em>flip<\/em>&#8221; one of the two letters, you must also flip the other, according to the condition on the elements of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-da3d7367d698288aefda3e09ffe11faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#85;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\"\/>).[\/vc_column_text][vc_column_text]<\/p>\n<h3>But how do you solve the puzzle?<\/h3>\n<p>[\/vc_column_text][vc_column_text]What is the maximum number of moves needed from a permissible arrangement of the exhibit to the initial state? This question is much more difficult to answer than the determination of the group <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/> generated by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/> above. First, let us give a (much too rough) upper bound for this: Each element of the group <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-2bfa5506aa3da9a7ba58450053dd6685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: 0px;\"\/> can be written as a concatenation of the rotations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-aeb1a518da71006d645faaa793ec45b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#122;&#44;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: -4px;\"\/> and their <a href=\"https:\/\/en.wikipedia.org\/wiki\/Inverse_element\" target=\"_blank\" rel=\"noopener\"><em>inverse<\/em><\/a>. Let us say that the &#8220;<em>elementary operations<\/em>&#8221; are exactly the rotations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-902e0599afa41ad7a412fb6b3a1c7e84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#120;&#94;&#50;&#44;&#120;&#94;&#51;&#44;&#121;&#44;&#121;&#94;&#50;&#44;&#121;&#94;&#51;&#44;&#122;&#44;&#122;&#94;&#50;&#44;&#122;&#94;&#51;&#44;&#119;&#44;&#119;&#94;&#50;&#44;&#119;&#94;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"277\" style=\"vertical-align: -4px;\"\/>.<\/p>\n<p>We now ask ourselves how many such elementary operations we have to perform at least one after the other in order to reach any element <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8563ce02dc8080c4916841d6a665e6fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#105;&#110;&#32;&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\"\/>.<\/p>\n<p>We know that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-15c3d98596deb564ae1c74093bad2f01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#71;&#124;&#61;&#50;&#94;&#56;&#92;&#99;&#100;&#111;&#116;&#32;&#57;&#33;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"88\" style=\"vertical-align: -5px;\"\/> holds (see above). With this we consider that at the latest after <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-e55c141399423ddd64267f465d62f5fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#124;&#71;&#124;&#45;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"\/>-many moves we can reach any permissible rearrangement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8563ce02dc8080c4916841d6a665e6fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#105;&#110;&#32;&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\"\/>, so until we can read the words &#8220;<em>Have the courage<\/em>&#8221; again. However, this number is <em>much much much<\/em> too large. for comparison: A <a href=\"https:\/\/en.wikipedia.org\/wiki\/Rubik%27s_Cube\" target=\"_blank\" rel=\"noopener\"><em>Rubik&#8217;s Cube<\/em><\/a> can always be transformed into its initial state in at most <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-421bce6278ed869a51558b9c9ab1a9b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\"\/> moves.<\/p>\n<p>Therefore, we now want to specify a lower bound that comes closer to reality: To do this, we consider how many permissible rearrangements we can obtain at most after performing exactly <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8ad02d5f0d297c7adacb6b8c4f9fee74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/>-many elementary operations (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-e89ff4d7bc07ab360a951b0cef96e331_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\"\/>). Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-e9783eadeed48e8a327455250cb98ad0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\"\/> be the number of these rearrangements. For <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-f67282e522f17e6f5c6ebbb54db1f306_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\"\/> we simply get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-fff1c35ed3e9eb2784f1533d3eb23456_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"48\" style=\"vertical-align: -3px;\"\/>, because we can, without doing anything, only produce the identical rearrangement. For <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-3297a83279057cbf74e3a4c32fb2b278_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\"\/> we get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-b82cff5631ccd37bfab6387f58aa0a42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#49;&#61;&#52;&#92;&#99;&#100;&#111;&#116;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"70\" style=\"vertical-align: -3px;\"\/>, because this is exactly the number of elementary operations and each of them yields a different rearrangement. Now suppose we knew <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-e9783eadeed48e8a327455250cb98ad0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\"\/> (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-4032bd8d24c31d18d888886bb8d2b6e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"40\" style=\"vertical-align: -3px;\"\/>). Let us say that we have performed exactly <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8ad02d5f0d297c7adacb6b8c4f9fee74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/> elementary operations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-902e0599afa41ad7a412fb6b3a1c7e84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#120;&#94;&#50;&#44;&#120;&#94;&#51;&#44;&#121;&#44;&#121;&#94;&#50;&#44;&#121;&#94;&#51;&#44;&#122;&#44;&#122;&#94;&#50;&#44;&#122;&#94;&#51;&#44;&#119;&#44;&#119;&#94;&#50;&#44;&#119;&#94;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"277\" style=\"vertical-align: -4px;\"\/> and that the last one was a multiple of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-82fc2d197e22b7961d3138638812f927_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>. Then it makes no sense to apply a multiple of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-82fc2d197e22b7961d3138638812f927_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> again, because this would give you a rearrangement that can already be achieved in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8ad02d5f0d297c7adacb6b8c4f9fee74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/> or less moves. So it can be assumed that then in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-cfc6a79b24dbcc26c7bbf5122bd25a04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#110;&#43;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"\/>-th move one of the elementary operations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-87f81899829acdc79626db2783bfda67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#44;&#121;&#94;&#50;&#44;&#121;&#94;&#51;&#44;&#122;&#44;&#122;&#94;&#50;&#44;&#122;&#94;&#51;&#44;&#119;&#44;&#119;&#94;&#50;&#44;&#119;&#94;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"209\" style=\"vertical-align: -4px;\"\/> is performed. Therefore, we get that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-4fb9312992626da60c98d4dd063fd935_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#92;&#108;&#101;&#113;&#32;&#57;&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"83\" style=\"vertical-align: -5px;\"\/>. Thus, for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-4032bd8d24c31d18d888886bb8d2b6e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"40\" style=\"vertical-align: -3px;\"\/> and the number of all configurations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-e1b2f824d7932e228ad2a6ae9c00bc26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\"\/> achievable in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8ad02d5f0d297c7adacb6b8c4f9fee74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/> or fewer moves, it follows that <\/p>\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 53px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-81f2a7f75e41c67f71e7ea565ef9bd31_l3.png\" height=\"53\" width=\"471\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#115;&#95;&#110;&#92;&#99;&#111;&#108;&#111;&#110;&#101;&#113;&#113;&#92;&#115;&#117;&#109;&#95;&#123;&#105;&#61;&#48;&#125;&#94;&#110;&#123;&#117;&#95;&#110;&#125;&#92;&#108;&#101;&#113;&#32;&#49;&#43;&#49;&#50;&#92;&#99;&#100;&#111;&#116;&#92;&#115;&#117;&#109;&#95;&#123;&#105;&#61;&#48;&#125;&#94;&#123;&#110;&#45;&#49;&#125;&#123;&#57;&#94;&#105;&#125;&#61;&#49;&#43;&#49;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#94;&#110;&#45;&#49;&#125;&#123;&#57;&#45;&#49;&#125;&#61;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#40;&#57;&#94;&#110;&#45;&#49;&#41;&#46;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>So what is the minimum number of moves needed to achieve each permissible rearrangement? Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-4032bd8d24c31d18d888886bb8d2b6e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"40\" style=\"vertical-align: -3px;\"\/> be the smallest number such that all elements <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8563ce02dc8080c4916841d6a665e6fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#105;&#110;&#32;&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\"\/> can be reached in at most <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8ad02d5f0d297c7adacb6b8c4f9fee74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/> moves. Then surely <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-952452af2c2c34daa9a224aa86e00980_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#95;&#110;&#61;&#124;&#71;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -5px;\"\/> must hold, and thus according to our calculation above <\/p>\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 36px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-dcea800abce2347992a9fb3df372c89a_l3.png\" height=\"36\" width=\"195\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#122;&#95;&#110;&#61;&#124;&#71;&#124;&#92;&#108;&#101;&#113;&#32;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#40;&#57;&#94;&#110;&#45;&#49;&#41;&#46;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>This leads to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-5eda22633ab7a4c7480e5b179dd70cb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#32;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"41\" style=\"vertical-align: -3px;\"\/>. That is much more realistic. But how big is the smallest integer <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-8ad02d5f0d297c7adacb6b8c4f9fee74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/> for the needed moves really? Find out for yourself by trying out the exhibit. Maybe you want to write a small computer programme that solves the problem?<\/p>\n<p>It makes sense to divide the problem into stages: First, rotate the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-07de1e9aebc049111b4109ae367f3ba6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"7\" style=\"vertical-align: 0px;\"\/> plate (as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/erlebnisland-mathematik.de\/wp-content\/ql-cache\/quicklatex.com-7885983a232453d9a4cb3ddae11da2bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#72;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\"\/>) to the correct position (top left). Then one solves the smaller remaining sub-problem and so on.[\/vc_column_text][vc_column_text]<\/p>\n<h3>Literature<\/h3>\n<p>[\/vc_column_text][vc_column_text][1] <a href=\"https:\/\/en.wikipedia.org\/wiki\/Symmetric_group\">https:\/\/en.wikipedia.org\/wiki\/Symmetric_group<\/a><\/p>\n<p>[2] <a href=\"https:\/\/en.wikipedia.org\/wiki\/Free_group\">https:\/\/en.wikipedia.org\/wiki\/Free_group<\/a><\/p>\n<p>[3] <a href=\"https:\/\/en.wikipedia.org\/wiki\/Cayley_graph\">https:\/\/en.wikipedia.org\/wiki\/Cayley_graph<\/a><\/p>\n<p>[4] <a href=\"https:\/\/en.wikipedia.org\/wiki\/Rubik%27s_Cube\">https:\/\/en.wikipedia.org\/wiki\/Rubik%27s_Cube<\/a><\/p>\n<p>[5] <a href=\"https:\/\/en.wikipedia.org\/wiki\/Wreath_product\">https:\/\/en.wikipedia.org\/wiki\/Wreath_product<\/a>[\/vc_column_text][\/vc_column][\/vc_row]<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>[vc_row drowwidth=&#8221;sidebar-biest-default sidebar-biest&#8221;][vc_column][vc_column_text] Have the courage Surely you have already worked with the Rubik&#8217;s Cube, or at least watched others do so. Of course, this cube has a lot to do with mathematics &#8212; but what exactly? The &#8220;Have the courage&#8221; puzzle is also of a similar nature. The commonality is that you can apply <a href=\"https:\/\/erlebnisland-mathematik.de\/en\/advanced-text-have-the-courage\/\" class=\"more-link\">&#8230;<span class=\"screen-reader-text\">  Advanced text Have the courage<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"folder":[],"class_list":["post-4159","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/4159","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/comments?post=4159"}],"version-history":[{"count":19,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/4159\/revisions"}],"predecessor-version":[{"id":4674,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/4159\/revisions\/4674"}],"wp:attachment":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/media?parent=4159"}],"wp:term":[{"taxonomy":"folder","embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/folder?post=4159"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}