Monday Math Madness #30 Winner: Placing Bath Tiles
24 Apr 2009 Daniel Sutoyo 6 comments 320 views
MMM #30 Winner
The winner for this edition of MMM is Ingmar Dasseville. Everyone was able to show that there is no way to place the bath tiles I have. Perhaps the problem was too easy and too similar to a classic chess problem. Thanks for everyone who submitted! One of the good explanation is provided below by Gareth McCaughan.
If you did not find this round of MMM to be challenging, next week is Sol’s turn over at wildaboutmath.com. Be prepared!
The Answer by Gareth McCaughan
The number of ways to place the tiles is 0.
Colour the squares of the board in checkerboard fashion. If two corners weren’t missing, there would be equal numbers of black and white squares. The two missing corners were the same colour, so there are two more (say) black squares than (say) white ones. But each 2×1 tile, however you place it, covers one white and one black square, so there is no way to use them to cover a set of squares with unequal numbers of black and white squares.
6 Responses to “Monday Math Madness #30 Winner: Placing Bath Tiles”
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I really don’t think this is a satisfactory explanation. You could tile the 10 by 10 board in such a way that those same corners were different colours (simply flip either of those two tiles), which would then (by your argument) suggest that since the missing corner squares were a white and a black, you could therefore tile the cutaway board somehow.
I agree that the answer is “no possible tiling”, but this explanation does not seem sound.
Sorry, I meant to clarify that I am aware of the allusion to the chess board problem, but with this tiling problem we are not bound to ‘checker’ the colours. That is, a valid tiling need not look like a checkerboard–it could have two adjacent squares of the same colour–and we cannot therefore make the assumption that begins the given explanation.
Nate, I think you misunderstood the assumption. You don’t place the tiles in a checkered fashion, but mark the floor on which you will put the tiles in a checkered fashion. This, you always can do. Every tile you will place, no matter its orientation or what the adjacent colours are, will cover a black and a white square.
[...] solution to MMM #30 is up at Blinkdagger. Here we go with [...]
Thanks, got it.