The teleporter drops Kinesys, with great precision, in the middle of the Abakos game hall. Here the locals spend their evenings counting combinations of dice, cards, and tiles, keeping score on abacuses the size of wardrobes.
On one table sits a chessboard with two corners cut off, next to a box of dominoes with one tile missing. An Abakian player challenges the crew: whoever covers the whole board wins a free recharge for the teleporter.
Liz gives it a determined try for half an hour. Dord refuses even to touch a “mutilated” chessboard. M00N, curious, starts looking at the colors of the squares, one by one…
The Riddle
The materials for this puzzle are a chessboard and 32 dominoes. Each domino is just the right size to cover exactly two adjacent squares of the board, so the 32 dominoes can cover all 64 squares.
Now suppose we remove the two squares at opposite ends of a diagonal, and also remove one domino. Is it possible to place the remaining 31 dominoes on the board so that they cover the 62 remaining squares? If so, show how; if not, prove that it can’t be done.
Hint
Before you try placing the dominoes, look at the colors: what color are the two squares you removed? And which squares does a domino always cover?
Solution
It is impossible to cover the reduced board with 31 dominoes, and it’s easy to prove.
The two diagonally opposite corners are the same color: removing them leaves a board with 32 squares of one color and only 30 of the other.
Every domino always covers two squares of different colors, because adjacent squares on a chessboard are always different colors. So any arrangement of 31 dominoes would cover 31 white squares and 31 black ones—but the remaining squares aren’t split that way. Put another way, after placing 30 dominoes you would always be left with two squares of the same color, which can’t be adjacent and so can’t be covered by the last domino.
M00N figured it out by looking at the colors. The Abakian player, it seems, knew all along.
Source: Martin Gardner, Mathematical Puzzles and Diversions (Italian edition: “Enigmi e giochi matematici”)