# 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

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---...+*
---*+O+*
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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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