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?

Label the squares as follows:

1 2 3 1 2 3 1 2
2 3 1 2 3 1 2 3
3 1 2 3 1 2 3 1
1 2 3 1 2 3 1 2
2 3 1 2 3 1 2 3
3 1 2 3 1 2 3 1
1 2 3 1 2 3 1 2
2 3 1 2 3 1 2 3

Each piece must cover a 1, a 2 and a 3.  There are 21 1's, 21 3's, but 22 2's, so the empty space must be on a 2.

Now start from the upper right corner:

2 1 3 2 1 3 2 1
3 2 1 3 2 1 3 2
1 3 2 1 3 2 1 3
2 1 3 2 1 3 2 1
3 2 1 3 2 1 3 2
1 3 2 1 3 2 1 3
2 1 3 2 1 3 2 1
3 2 1 3 2 1 3 2

The same statistics apply to this and so the empty square must again be on a 2.  The only places that are a 2 in both diagrams are as Joe specified, or in numeric coordinates, (3,3), (3,6), (6,3) and (6,6).

One such covering can be obtained, first by placing a 2-unit ring around the edge:

A B C C C D D D
A B E E E F F F
A B         G H
I J         G H
I J         G H
I J         K L
M M M N N N K L
O O O P P P K L

Then the center:

  Q Q Q
R S S S
R T T T
R U U U

Posted by Charlie on 2005-11-12 13:53:23