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Topic: Another Tiling Problem - But Even Cooler (Read 636 times) |
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william wu
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Another Tiling Problem - But Even Cooler
« on: Sep 25th, 2003, 12:43am » |
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Another tiling problem. This time more different from the others though, I think.
Consider a gridded board of length 2n units and width 2n units, where n is any positive integer. Exactly one of the squares on this board is missing, although you don't know which square.
You have L-shaped tiles that consist of three unit side length squares: one for each leg of the L, and one for the joint. Can you cover the board with these tiles? (You don't want to cover the missing square.) Sometimes? Always? Never? Prove it.
[edit]12:50 PM 9/28/2003 Added that the squares of the L-shaped tiles are of unit side length.
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| « Last Edit: Sep 28th, 2003, 12:51pm by william wu » |
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towr
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Re: Yet Another Tiling Problem
« Reply #1 on: Sep 25th, 2003, 3:45am » |
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Yes, allways. Provable by induction. (but I won't give it away)
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James Fingas
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Re: Yet Another Tiling Problem
« Reply #2 on: Sep 25th, 2003, 5:25am » |
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That's a very cool puzzle.
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william wu
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Re: Another Tiling Problem - But Even Cooler
« Reply #3 on: Sep 25th, 2003, 5:53pm » |
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Yup, pretty cool; also a good way to introduce people to a neat approach for solving problems, called ...
hint: :: divide and conquer ::
[edit] 1:35 AM 10/22/2003 added this hint
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| « Last Edit: Oct 22nd, 2003, 1:35am by william wu » |
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Margit Schubert-While
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Re: Another Tiling Problem - But Even Cooler
« Reply #4 on: Sep 28th, 2003, 4:18am » |
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Well, I would say that the answer to the problem as stated is indeterminate. There is NO definition of the sizes of the 3 squares that make the tile. Now if you state that the size of the squares is related to "n", then that's another matter.
Margit
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william wu
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Re: Another Tiling Problem - But Even Cooler
« Reply #5 on: Sep 28th, 2003, 12:55pm » |
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Added that the squares which compose the L-shaped tiles are of unit side length; let me know if there's anything else I can improve
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Barukh
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Re: Another Tiling Problem - But Even Cooler
« Reply #6 on: Sep 28th, 2003, 11:54pm » |
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[smiley=blacksquare.gif]
1. Build a sequence of L-shapes L1, L2, ..., Ln, where Li is a shape 2i-1 units wide. 4 Li-s may be combined to form an Li+1.
2. Start tiling by putting Ln on a board so that the quadrant with the hole is left uncovered.
[smiley=blacksquare.gif]
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