91Ó°ÊÓ

Show that an \(m\) -by-n chessboard has a perfect cover by dominoes if and only if at least one of \(m\) and \(n\) is even.

Imagine a prison consisting of 64 cells arranged like the squares of an 8 -by-8 chessboard. There are doors between all adjoining cells. A prisoner in one of the corner cells is told that he will be released, provided he can get into the diagonally opposite corner cell after passing through every other cell exactly once. Can the prisoner obtain his freedom?

Find the number of different perfect covers of a 3 -by- 4 chessboard by dominoes.

Verify that there is no magic square of order 2 .

Use de la Loubère's method to construct a magic square of order \(7 .\)

Use de la Loubère's method to construct a magic square of order \(9 .\)

Show that a magic square of order 3 must have a 5 in the middle position. Deduce that there are exactly 8 magic squares of order \(3 .\)

Show that there is no magic cube of order \(2 .\)

Construct a pair of orthogonal Latin squares of order \(4 .\)

A 6-by-6 chessboard is perfectly covered with 18 dominoes. Prove that it is possible to cut it either horizontally or vertically into two nonempty pieces without cutting through a domino; that is, prove that there must be a fault line.