{"id":4329,"date":"2022-09-12T12:42:57","date_gmt":"2022-09-12T10:42:57","guid":{"rendered":"https:\/\/erlebnisland-mathematik.de\/?page_id=4329"},"modified":"2023-01-26T11:32:59","modified_gmt":"2023-01-26T10:32:59","slug":"advanced-text-klarners-theorem","status":"publish","type":"page","link":"https:\/\/erlebnisland-mathematik.de\/en\/advanced-text-klarners-theorem\/","title":{"rendered":"Advanced text Klarner’s theorem"},"content":{"rendered":"

[vc_row drowwidth=”sidebar-biest-default sidebar-biest”][vc_column][vc_column_text]<\/p>\n

Klarner’s theorem<\/h1>\n

How do I fit as many boxes and suitcases as possible into my much too small car? How are container ships loaded in the most space-saving way possible? How can I cut out as many biscuits as possible from a sheet of dough? How does the layout artist fill a newspaper page with as many advertisements as possible without gaps?<\/em><\/p>\n

Such questions, which play a major role in everyday life and in business, are what mathematicians call packing problems<\/em>. They were a speciality of the American mathematician David A. Klarner<\/em> (\u20201999).<\/p>\n

A puzzle in the Maths Adventure Land refers to one of its central statements. Klarner’s theorem<\/em> states: A rectangle of dimension \"a\times b\", which consists of squares with side length 1, can be filled without gaps with \"1\times n\"-sized strips if at least one of the side lengths \"a\" or \"b\" is divisible by \"n\". The rectangle in the Maths Adventure Land is divided into \"10\times 10\", or \"100\" squares. The strips with which this area is to be filled consist of four squares of the same size arranged in a row one behind the other. \"25\" of these strips also make \"100\" squares.[\/vc_column_text][vc_column_text]However, the task of completely covering the rectangle with the strips is not solvable. If \"24\" strips are placed on the surface, in no case a row of \"4\" squares remains free for the 25th strip, but always a square field of \"2\times 2\" small squares. Klarner’s theorem is thus confirmed, because the side length \"10\" is not divisible by the strip length \"4\".[\/vc_column_text][vc_single_image image=”1269″ img_size=”large” alignment=”center”]Figure 1: Exhibit in the Maths Adventure Land[\/vc_single_image][vc_column_text]<\/p>\n

And now … the mathematics:<\/h3>\n

[\/vc_column_text][vc_column_text]The divisibility<\/em> of the large area of \"10\times 10\" squares can be well illustrated if the diagonals are marked with four different colours. In the Maths Adventure Land, the 10 squares of the main diagonal from the bottom left to the top right are coloured red<\/em>, the squares of the side diagonals connecting to the bottom right are coloured blue<\/em>, orange<\/em>, green<\/em> and red<\/em> again until the blue<\/em> square in the corner. In reverse order, the secondary diagonals are coloured towards the top left, i.e. green<\/em>, orange<\/em>, blue<\/em> and red<\/em> again and so on until the green<\/em> square in the corner (cf. Figure 2). In mathematical terms, if one identifies each square with a grid point<\/em> \"(x,y)\" in the plane grid \"\mathbb Z\times\mathbb Z\", those grid points where \"x-y\" belong to the same residue class modulo<\/em> \"4\" are coloured with the same colour.<\/p>\n

If you now distribute the strips of four in any order, each strip always covers one square of each of the four colours above. If you now add up all the squares of each colour, you get<\/p>\n