#Tessellation proof problem

22 messages · Page 1 of 1 (latest)

bright cedar
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A tessellation (or tiling) is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. It is clear that any 2n × 2n square can be tiled by 2 × 1 rectangular tiles (called dominoes) in a number of different patterns. Let us say that a tiling is interlocking if the 2n × 2n square cannot be partitioned into two rectangles that freely slide along each other. Show that the smallest square that admits an interlocking tiling with dominoes is the 8 × 8 square.

I can find a tiling so that the 8 x 8 fails but I don't know how to go about proving that it is the smallest case? It's pretty easy to show why the 2 x 2 and 4 x 4 works, but I'm not sure how to go about showing the 6 x 6 case always working.

lucid lanceBOT
drifting whale
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can you show the 8x8 pattern you got?

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tends to help

bright cedar
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blue are vertical tiles and red are horizontal. each 2x1 tile has the same number in each tile

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im sure there are quite a few ways to tile the 8x8 grid

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maybe it has something to do with symmetry about the diagonal?

drifting whale
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My guess it's that it's related to the fact there are 10 ways to cut a 6x6 triangle and 18 dominoes.

The rest of the messages are the answer, I spoilered them for you. The above text is a good hint

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|| to recieve an interlocked pattern every line that can cut the rectangle must have a domino straddling it||

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||however, it must have atleast 2 dominoes, since having one straddling it would leave an odd amount of squares in one part of the rectangle, which is impossible to tile||

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|| thus there are atleast 10 dominoes that straddle a line, and atleast another 10 that are "parallel" to the previous ones and don't intersect a new line that hasn't been intersected before||

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|| but there are only 18 dominoes. So an interlocked tiling is impossible||

bright cedar
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Ok thanks, I'll try to figure it out without the spoilers

bright cedar
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Wait nevermind

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:o

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this is great, thank you!

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that is cool wow

drifting whale
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There is a great question this reminded me of

jagged kestrel
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.solved