Mutilated chessboard problem

The mutilated chessboard problem is a tiling puzzle proposed by philosopher Max Black in his book Critical Thinking (1946). It was later discussed by Solomon W. Golomb (1954), Gamow & Stern (1958) and by Martin Gardner in his Scientific American column "Mathematical Games". The problem is as follows:

Suppose a standard 8×8 chessboard has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2×1 so as to cover all of these squares?

Mutilated chessboard problem

Most considerations of this problem in literature provide solutions "in the conceptual sense" without proofs.[1] John McCarthy proposed it as a hard problem for automated proof systems.[2] In fact, its solution using the resolution system of inference is exponentially hard.[3]

Solution
Mutilated chessboard example showing like colored empty squares (note the two empty black squares at the middle)

The puzzle is impossible to complete. A domino placed on the chessboard will always cover one white square and one black square. Therefore, a collection of dominoes placed on the board will cover an equal numbers of squares of each color. If the two white corners are removed from the board then 30 white squares and 32 black squares remain to be covered by dominoes, so this is impossible. If the two black corners are removed instead, then 32 white squares and 30 black squares remain, so it is again impossible.[4]

Analogue

A similar problem asks if an ant starting at a corner square of an unmutilated chessboard can visit every square exactly once, and finish at the opposite corner square. Diagonal moves are disallowed; each move must be along a rank or file.

Using the same reasoning, this task is impossible. For example, if the initial square is white, as each move alternates between black and white squares, the final square of any complete tour is black. However, the opposite corner square is white.[5]

Gomory's theorem
A
B
Removing one black and one white square on this Hamiltonian cycle (examples shown with ×) yields one (A) or two (B) paths with even numbers of squares

The same impossibility proof shows that no domino tiling exists whenever any two white squares are removed from the chessboard. However, if two squares of opposite colors are removed, then it is always possible to tile the remaining board with dominoes; this result is called Gomory's theorem,[6] and is named after mathematician Ralph E. Gomory, whose proof was published in 1973.[7] Gomory's theorem can be proven using a Hamiltonian cycle of the grid graph formed by the chessboard squares; the removal of two oppositely colored squares splits this cycle into two paths with an even number of squares each, both of which are easy to partition into dominoes.

See also

Notes
  1. Andrews, Peter B.; Bishop, Matthew (1996), "On Sets, Types, Fixed Points, and Checkerboards", Theorem Proving With Analytic Tableaux and Related Methods: 5th International Workshop, Tableaux '96, Terrasini, Palermo, Italy, 15-17th, 1996, Proceedings, Lecture Notes in Computer Science, Springer-Verlag, most treatments of the problem in the literature solve it in the conceptual sense, but do not actually provide proofs of the theorem in either of McCarthy's original formulations.
  2. Arthan, R. D. (2005), The Mutilated Chessboard Theorem in Z (PDF), retrieved 2007-05-06, The mutilated chessboard theorem was proposed over 40 years ago by John McCarthy as a “tough nut to crack” for automated reasoning.
  3. Alekhnovich, Michael (2004), "Mutilated chessboard problem is exponentially hard for resolution", Theoretical Computer Science, 310 (1–3): 513–525, doi:10.1016/S0304-3975(03)00395-5.
  4. McCarthy, John (1999), "Creative Solutions to Problems", AISB Workshop on Artificial Intelligence and Creativity, retrieved 2007-04-27
  5. Miodrag Petkovic, Mathematics and Chess: 110 Entertaining Problems and Solutions, ISBN 0-486-29432-3
  6. Watkins, John J. (2004), Across the board: the mathematics of chessboard problems, Princeton University Press, pp. 12–14, ISBN 978-0-691-11503-0.
  7. According to Mendelsohn, the original publication is in Honsberger's book. Mendelsohn, N. S. (2004), "Tiling with dominoes", The College Mathematics Journal, Mathematical Association of America, 35 (2): 115–120, doi:10.2307/4146865, JSTOR 4146865; Honsberger, R. (1973), Mathematical Gems I, Mathematical Association of America.

References
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