63rd Czech and Slovak Mathematical Olympiad

Question

Let us call by an "edge" any segment of length

1

which is common to two adjacent fields of a given chessboard

8 \times 8

. Consider all possible cuttings of the chessboard into

32

pieces

2 \times 1

and denote by

n (e)

the total number of such cuttings that involve the given edge

e

. Determine the last digit of the sum of the numbers

n (e)

over all the edges

e

.

(Michal Rolínek)

Accepted Answer

The number of edges, which are not involved in a given cutting, is equal to 32, because each of these edges must coincide with the common segment of the two fields forming one of the 32 resulting pieces 2 1. Thus each cutting gives a contribution 112 - 32 = 80 to the sum S of all the numbers n(e). Consequently, the sum S is a multiple of 80 and thus its last digit is zero.