168 geometry/coloring/triominoes.p
Description
This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.
168 geometry/coloring/triominoes.p
There is a chess board (of course with 64 squares). You are given 21
"triominoes" of size 3-by-1 (the size of an individual square on a
chess board is 1-by-1). Which square on the chess board can you cut out
so that the 21 triominoes exactly cover the remaining 63 squares? Or is
it impossible?
geometry/coloring/triominoes.s
|||||||| |||||||| |||||||| ---***+* ---...+* ---*+O+* ---*+... ---*+***
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:
ABCABCAB CABCABCA BCABCABC ABCABCAB CABCABCA BCABCABC ABCABCAB CABCABCA
Every triomino 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:
CABCABCA ABCABCAB BCABCABC CABCABCA ABCABCAB BCABCABC CABCABCA ABCABCAB
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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