Problem:
On a chessboard, a king moves according to the following rules:
- It can move one square at a time, horizontally, vertically, or diagonally. (These are the usual moves of the king in chess.)
- It can move in each of the eight allowable directions at most three times in its entire route.
The king can start at any square. Determine
(a) whether the king can visit every square;
(b) whether the king can visit every square except the center.
Solution
Solution:
a. The answer is no. To visit all 25 squares, the king must make his maximum of 24 moves and thus must move in each of the eight allowable directions exactly 3 times. Three of these directions (or 9 moves) take the king from a row to the next higher row; three directions (or 9 moves) go to the next lower row, and the remaining two directions and six moves keep the king in the same row. Because the numbers of upward and downward moves are equal, the king must start and end in the same row. Similarly, he must start and end in the same column and thus in the same square. Thus the 25th square visited will repeat the first and at least one square will be missed.
b. The answer is yes. A solution is shown. Trickily, all known solutions are completely asymmetric.
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