8
$\begingroup$

Jepetto the toymaker was thinking about a new toy to add to his tiling product line. His new design involved a punctured chessboard: an ordinary $8 \times 8$ chessboard, except with a single square removed.

Now, when he ordered these punctured chessboards, Jepetto was very specific about the missing square; so he was furious when the manufacturer showed up with a chessboard whose hole was in the wrong place. Jepetto yelled at the man:

Don't you see? You've ruined it! How can anyone tile this board with $3\times 1$ rectangular pieces!?

On how many squares could the hole have been misplaced?

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  • $\begingroup$ What exactly is tiling with rectangular pieces? Can they be different? Can they be square? $\endgroup$ – xnor Mar 22 '16 at 5:52
  • $\begingroup$ @xnor They are all $3 \times 1$ rectangles. $\endgroup$ – Fimpellizieri Mar 22 '16 at 6:13
  • $\begingroup$ Hah, apparently my phone doesn't render MathJax. I was confused seeing your apparent response "They are all rectangles." $\endgroup$ – xnor Mar 22 '16 at 6:26
  • $\begingroup$ @MikeEarnest Oh, indeed a duplicate. :( $\endgroup$ – Fimpellizieri Mar 22 '16 at 17:55
7
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If we 3-color the chessboard as

ABCABCAB
BCABCABC 
CABCABCA
ABCABCAB
BCABCABC
CABCABCA
ABCABCAB
BCABCABC

we see that any 1x3 rectangle covers exactly one of each of A, B, C. Since there's 21 A's, 22 B's, and 21 C's, the missing square must be a B. But, the same would be true if we reflected or rotated the coloring, in which case the only squares that are always B's are:

........
........
..B..B..
........
........
..B..B..
........
........

So, no other square can be removed, and we can indeed tile the board with one of these squares missing, giving us 4 possible removable squares

|–––|–––
|–––|–––
|–––|B||
––––––||
|–––||||
|–––||||
|–––||||
––––––||
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  • 1
    $\begingroup$ While you appear to have the logic correct, you did not actually state an answer to the question: "On how many squares could the hole have been misplaced?" We could easily do the simple math ourselves, based on your logic. But, it would be nice if you did state the actual answer. Spoiler tags would also be considerate. $\endgroup$ – Makyen Mar 22 '16 at 10:46

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