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 Covering a chessboard (Posted on 2005-11-12)
You are given 21 3x1 rectangular pieces to cover an 8x8 chessboard. Since the board has 64 squares, which square on the chessboard must you cut out so that the 21 given pieces exactly cover the remaining 63 squares? Or it is impossible, no matter which square you remove?

 Submitted by pcbouhid Rating: 3.6667 (6 votes) Solution: (Hide) There is only one way to remove a square, aside from rotations and reflections. To see that there is at most one way, do this: Label all the squares of the chessboard with A, B or C in sequence by rows starting from the top:``` A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A``` Every piece must cover one A, one B and one C. There is one extra A square, so an A must be removed. Now label the board again by rows starting from the bottom:``` C A B C A B C A A B C A B C A B B C A B C A B C C A B C A B C A A B C A B C A B B C A B C A B C C A B C A B C A A B C A B C A B``` The square removed must still be an A. The only squares that got marked with A both times are these:``` . . . . . . . . . . . . . . . . . . A . . A . . . . . . . . . . . . . . . . . . . . A . . A . . . . . . . . . . . . . . . . . .```

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 Subject Author Date re: Count the Ways Charlie 2005-11-14 08:24:58 re: Count the Ways Charlie 2005-11-13 19:22:35 Count the Ways Charlie 2005-11-13 19:18:14 re(6): spoiler Mindy Rodriguez 2005-11-13 12:49:01 A solution Ken Haley 2005-11-13 08:14:48 re(6): spoiler Vernon Lewis 2005-11-13 08:07:17 re(5): spoiler pcbouhid 2005-11-13 05:21:43 re(4): spoiler Mindy Rodriguez 2005-11-12 22:37:31 re(3): spoiler pcbouhid 2005-11-12 21:54:42 re(2): spoiler Mindy Rodriguez 2005-11-12 21:28:54 re: spoiler pcbouhid 2005-11-12 20:27:42 spoiler Mindy Rodriguez 2005-11-12 18:08:40 early thoughts. Vernon Lewis 2005-11-12 14:40:32 solution Charlie 2005-11-12 13:53:23 My thoughts: Joe 2005-11-12 11:03:45
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