 
Summary: Mutilated Chessboard problem is exponentially hard for
Resolution
Michael Alekhnovich
October 9, 2002
Abstract
Mutilated Chessboard principle CB n says that it is impossible to cover by domino
tiles the chessboard 2n 2n with two diagonally opposite corners removed. We prove
2
p
n) lower bound on the size of minimal Resolution refutation of CB n .
1 Introduction
Mutilated Chessboard problem corresponds to the wellknown puzzle that often appears in
mathematical popular literature. It asks to cover by dominos a 2n2nchessboard with two
white squares removed from the opposite corners. The goal is well known to be unachievable
because every domino tile covers exactly one white and one black square while the mutilated
chessboard contains two white squares less.
This principle is probably one of the earliest proposed hard problems for theorem provers.
Expressed as a tautology in rst order logic, it was proposed as a \tough nut for proof
procedures" by McCarthy in 1964 [M64]. It was considered again by Krishnamurthy in 1985
