Magic in a Square

N² numbers are arranged in a square pattern. Select one of the numbers and try answering computer queries. After two attempts, computer will reveal your selected number.

Explanation

Copyright © 1996-2009 Alexander Bogomolny

Simple as the trick is, it was described by [Rouse Ball, p. 325] and referred to by [M. Gardner, p. 3]. Martin Gardner writes: "The success of the trick depends, of course, on the spectator's inability to follow the procedure well enough to guess the operating principle." He then selfishly adds: "Unfortunately, few spectators are that dense."

Every location in the square is determined by its row and column numbers. The column is indicated directly on the first response, after which the columns of the square are swapped with the corresponding rows. Answer to the second query therefore determines the row. To confound the mildly dense spectators, computer, before displaying the numbers, randomly reshuffles them in every column.

References

1. M. Gardner, Mathematics Magic and Mystery, Dover, 1956
2. W. W. Rouse Ball, H. S. M. Coxeter, Mathematical Recreations and Essays, Dover, 1987

Copyright © 1996-2009 Alexander Bogomolny