Puzzles based on the chessboard have been popular since the medieval era, when chess itself first became widely popular. In recent times, a puzzle genre has emerged revolving around the rules of chess, as seen in the collection by the late Raymond Smullyan, *The Chess Mysteries of Sherlock Holmes *(1979), where the puzzles are framed as dialogues between master fictional sleuth Sherlock Holmes and his assistant Watson: “Black moved last, Watson. What was his last move—and White’s last move?” Without giving away the answer, which I leave up to the reader, suffice it to say that there is a trick here—a piece (like a White Bishop) was blocking the attack, and White moved another piece (like a Rook) out of the way, resulting in a discovered check.

Solving such puzzles involves an integrated use of spatial reasoning (determining where the pieces are on the board and how they can move about) and insight thinking (discovering the trick via an “Aha!” insight). To put it colloquially, in such puzzles, spatial reasoning “meets” insight thinking. Some chessboard-based puzzles, however, might enlist these two processes in an entangled way instead, with the insight resulting from disentangling the puzzle from how it describes a situation spatially.

An Example of a Chessboard-Based Puzzle

A classic example is the so-called mutilated chessboard problem, invented by philosopher Max Black in his 1946 book Critical Thinking. It can be paraphrased as follows:

If two opposite corners of a standard 8-by-8 checkerboard are removed, can the checkerboard be covered by dominoes? Assume that the size of each domino is the size of two adjacent squares of the checkerboard. The dominoes cannot be placed on top of each other and must lie flat.

It is not possible to cover the altered checkerboard in this way, for the reason that the two squares that are removed will be of the same color, either two white or two black (as you can confirm on a standard chessboard). That is the key insight. Each domino, no matter how it is placed, must cover every adjacent black and white square, which is not possible on a board with an unequal number of colored squares.

The puzzle highlights how spatial and insight thinking can become intertwined. The solution rests in bypassing any brute-force attempt at spatial tiling. It reveals, in effect, how our inbuilt ingenuity allows us to overcome mental blockages. The puzzles featured in this post are classic ones in this genre (including versions). They reveal how the two modes of thinking are sometimes used in tandem and sometimes in an entangled way—with one or the other mode dominating according to the puzzle.

Additional Puzzles

1. Here is a version of the mutilated chessboard problem. Remove the four corners of a regular 8-by-8 chessboard. Can the board now be covered by the dominoes?

2. Here is another variant. Remove the two adjacent squares (white-black, black-white) from all four corners. Can the board be covered by the dominoes?

3. In 1256, the Arab jurist Ibn Khallikan formulated a now classic puzzle, which can be paraphrased as follows. How many grains of wheat are needed on the last square of a 64-square chessboard if 1 grain is to be put on the first square of the board, 2 on the second, 4 on the third, 8 on the fourth, and so on?

4. Here is a version. The grains that are put on each white square increase by one: 1 on the first white square, 2 on the next white square, 3 on the one after, and so on. On each of the black squares, only one grain is put. How many grains does the chessboard hold?

5. How many total square configurations of all sizes, from 1-by-1 to 8-by-8, exist on a standard chessboard?

6. Each white square is replaced with the number 1 and each black square with the number 2. Summing up the numbers, 32 (white) + 64 (black), we get 96. What is the least number of white and/or black squares that must be removed so that the total adds up to 92?

7. Here is a well-known chess riddle, just for a change. Two chess masters played five games of chess, leaving with three wins, no losses, and no draws. How is this possible?

Answers

1. Yes, since there are two opposite white corners and two opposite black corners. Removing these leaves 60 squares, consisting of 30 adjacent black-white (or white-black) squares, and thus coverable by the dominoes.

2. Yes, since two white-black and two black-white corners are removed, leaving 56 squares, consisting of 28 adjacent black-white or white-black squares, and thus coverable by the dominoes.

3. The number of grains on the first square can be represented with 20, which equals 1, the number on the second square by 21, which equals 2, the number on the third square by 22, which equals four, and so on. There are 64 squares, so the number of grains on each successive square of the chessboard can be represented by the series {20, 21, 22, 23, 24, …, 263}. The last number is so enormous that it would require considerable effort to compute it. We’ll leave it at that.

4. There are 32 white squares, and the number of grains on these is the sum of the first 32 numbers: 1 + 2 + 3 + 4 + … + 32 = 528. There are 32 black squares with one grain on each one, which is 32 in total. Adding these up, 528 + 32, we get 560.

5. Locate each size of square across the board. A square of size 2-by-2 (consisting of two black and two white constituent squares), for example, can be found in 7 locations horizontally and 7 locations vertically, for 49 different positions in total. Overall: number of 1-by-1 squares (each single square on the board) = 64; number of 2-by-2 squares = 49, number of 3-by-3 squares = 36, number of 4-by-4 squares = 25, number of 5-by-5 squares = 16, number of 6-by-6 squares = 9, number of 7-by-7 squares = 4, number of 8-by-8 squares = 1 (the whole board). Total: 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squares.

6. 92 is four less than 96. The least number of squares that sum to 4, which are to be removed from the board, is two black squares.

7. They played against different opponents, not against each other.