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A TOUGH NUT FOR PROOF
Computer Science Department
Stanford, CA 94305
1999 Jan 2, 5:30 p.m.
STANFORD ARTIFICIAL INTELLIGENCE PROJECT
Memo No. 16, July 17, 1964
It is well known to be impossible to tile with dominoes a checker-
board 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 pred-
icate 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-
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
11999: Life was simpler when AI was supported directly out of the Oﬃce of the Secre-
tary of Defense.
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 tableaus1, into a
resolvent argument2, or into a standard proof. Therefore, I oﬀer the problem
of proving the following sentences inconsistent as a challenge to the program-
mers 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
We number the rows and columns from 1 to 8 and we introduce predicates
S(x,y), L(x,y), E(x,y), G1(x, y), G2(x, y), G3(x, y), G4(x, y), and G5(x, y)
with the following intended interpretations: S(x,y) means y = x + 1
L(x,y) means x < y
E(x,y) means x = y
G1(x, y) means the square (x,y) and the square (x+1,y) are covered by a
G2(x, y) means the square (x,y) and the square (x,y+1) are covered by a
G3(x, y) means the square (x,y) and the square (x-1,y) are covered by a
G4(x, y) means the square (x,y) and the square (x,y-1) are covered by a
G5(x, y) 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.
the values of S(x,y), L(x,y), and E(x,y) for x, y = 1, · · ·, 8.
11. G5(1, 1) ∧ G5(8, 8)
These axioms insure that every square (x,y) satisﬁes exactly one
12. G5(x, y) ⊃ (E(1, x) ∧ E(1, y)) ∨ (E(8, x) ∧ E(8, y)) These axioms insure
that the uncovered squares are precisely (1,1) and (8,8).
13. S(x1, x2) ⊃ G(x1, y) ≡ G3(x2, y)
14. S(y1, y2) ⊃ G2(x, y1) ≡ G4(x, y2) These axioms state the conditions
that a pair of adjacent squares be covered by a domino.
15. ¬G3(1, y) ∧ ¬G1(8, y) ∧ ¬G2(x.8) ∧ ¬G4(x, 1)
These axioms state that the dominoes don’t stick out over the edge of
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 indi-
viduals 1(cid:48), . . . , 8(cid:48) satisfying the relations asserted for 1, . . . , 8 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 G1(x, y) or G2(x, y) 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
In a formalism that allows functions and equality we have a briefer in-
consistent set of sentences involving
the successor of x
g(x, y) has value of 1 to 5 according to whether G1(x, y) or · · · or G5(x, y)
The sentences are
s(s(s(s(x))))= x The sentences insure the existence of 8 distinct individuals using a cyclic successor function.
g(x,y)= 5 ≡ x = 8 ∧ y = 8 ∨ x = 1 ∧ y = 1 Insures that exactly the corner squares (1,1) and (8,8) are uncovered.
g(x,y)= 1 ≡
g(x,y)= 2 ≡
g(x,s(y))= 4 Each domino covers two adjacent squares
g(1, y)(cid:54)= 3 ∧
g(8, y)(cid:54)= 1 ∧
g(x, 1)(cid:54)= 4 ∧
g(x, 8)(cid:54)= 2 Dominoes don’t stick out
s(8)∧ 2 =
s(1)∧ 3 =
s(2)∧ 4 =
s(3)∧ 5 =
g(x,y)= 1 ∨
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.
/@steam.stanford.edu:/u/jmc/f91/toughnut.tex: begun Sat Jan 2 13:51:51 1999, latexed January 2, 1999 at 5:30 p.m.