John McCarthy
Computer Science Department
Stanford University
Stanford, CA 94305
jmc@cs.stanford.edu
http://www-formal.stanford.edu/jmc/
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It is well known to be impossible to tile with dominoes a checkerboard with two opposite corners deleted. This fact is readily stated in the first order predicate calculus, but the usual proof which involves a parity and counting argument does not readily translate into predicate calculus. We conjecture that this problem will be very difficult for programmed proof procedures.
The research reported here was supported in part by the Advanced
Research Project Agency of the Office of the Secretary of Defense (SD-183).
It is impossible to cover the mutilated checkerboard shown in the
figure with dominoes like the one in the figure. Namely, a domino
covers a square of each color, but there are 30 black squares and 32
white squares to be covered.
This old impossibility statement is readily formulated as a sentence
of the predicate calculus, but I don't see how the parity and counting
argument can be translated into a guide to the method of semantic
tableaus 
, into a resolvent argument 
, or into a standard proof.
Therefore, I offer the problem of proving the following sentences
inconsistent as a challenge to the programmers of proof procedures
and to the optimists who believe that by formulating number theory in
predicate calculus and by devising efficient general proof procedures
for predicate calculus, significant mathematical theorems can be
proved.
We number the rows and columns from 1 to 8 and we introduce predicates
S(x,y), L(x,y), E(x,y), 
, 
, 
,
, and 
with the following intended interpretations:
S(x,y) means y = x + 1
L(x,y) means x < y
E(x,y) means x = y
means the square (x,y) and the square (x+1,y) are
covered by a domino.
means the square (x,y) and the square (x,y+1) are covered
by a domino.
means the square (x,y) and the square (x-1,y) are covered
by a domino.
means the square (x,y) and the square (x,y-1) are covered
by a domino.
means the square (x,y) is not covered.
We shall axiomatize only as much of the properties of the numbers from 1 to 8 as we shall need.
S(2,3) S(3,4) S(4,5) S(5,6)
S(6,7) S(7,8)
L(x,y) Lyz Lxz 
S(x,z) E(x,y)
These axioms insure that all eight numbers are
different and determine the values of S(x,y), L(x,y), and E(x,y) for x, y =
1, 
, 8.
These
axioms insure that every square (x,y) satisfies exactly one 
These axioms insure that the uncovered squares are
precisely (1,1) and (8,8).
These axioms state the conditions that a pair of adjacent
squares be covered by a domino.
These axioms state that the dominoes don't stick out over the edge of the board.
Suppose we had a model of these 15 sentences (in Robinson's
clausal formalism, there would be 31 clauses). There would have to be eight
individuals 
satisfying the relations asserted for
in the axioms. They would have to be distinct since
axioms 1,2, and 3 allow us to prove L(x,y) whenever this is so and axioms
4 and 5 then allow us to show that L(x,y) holds only for distinct x and
y.
We then label the squares of a checkboard and place a domino on each
square (x,y) that satisfies or sticking to
the right or up as the case may be. Axioms 13 and 14 insure that the
dominoes don't overlap, axioms 6-12 insure that all squares but the
corner squares are covered and axiom 15 insures that no dominoes stick
out over the edge.
Since there is no such covering the sentences have no model and are inconsistent.
In a formalism that allows functions and equality we have a briefer inconsistent set of sentences involving
s(x)the successor of x
g(x,y) has value of 1 to 5 according to whether
or or
The sentences are
The sentences insure the existence of 8 distinct individuals using a cyclic successor function.
x = 8 y = 8 
x = 1
y = 1 Insures that exactly the corner squares (1,1) and (8,8) are uncovered.
g(s(x),y) = 3 g(x,s(y)) = 4 Each domino covers two adjacent squares
Dominoes don't stick out
2 = s(1) 3 = s(2) 4 =
s(3) 5 = s(4) g(x,y) = 2 g(x,y) = 3 g(x,y)
= 4 g(x,y) = 5
Identifies the numbers used and ties down the values of g.