Notes on the "15" Puzzle

the 16th square being left blank. Fifteen counters, numbered in like manner, are placed at random upon the sqaares so that one square is vacant. The counter occupying any adjacent square may now be moved into the vacant square-thus: If No. 7 is vacant, either of the counters occupying Nos. 3, 6, 8, 11 can be moved into it, but no diagonal move is allowed. The puzzle is to bring all the counters into their proper squares by successive moves. It seems to be generally supposed, by those who have tried the puzzle, that this is always possible, whatever be the original random position of the counters, but this is an error, as the following demonstration will show: When the blank or sixteenth square is the vacant one, the arrangement of the counters may be called a positive or negative one, according as the term of the 15-square determinant, which has for first anid second subscripts the numberson the squares and counters, is positive or negative. Let n 101 397