By asking a small number of innocent-sounding questions about an unknown number, it is possible to reconstruct the number with absolute certainty (assuming that the questions are answered correctly). Ball and Coxeter (1987) give a number of sets of questions which can be used.

One of the simplest algorithms uses only three questions to determine an unknown number :

- 1. Triple and announce if the result is Even or Odd.
- 2. If you were told that is Even, ask the person to reveal the number which is half of . If you were told that is Odd, ask the person to reveal the number which is half of .
- 3. Ask the person to reveal the number of times which 9 divides evenly into .

Another method asks:

- 1. Multiply the number by 5.
- 2. Add 6 to the product.
- 3. Multiply the sum by 4.
- 4. Add 9 to the product.
- 5. Multiply the sum by 5 and reveal the result .

**References**

Bachet, C. G. *Problèmes plaisans et délectables, 2nd ed.* 1624.

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations and Essays, 13th ed.* New York: Dover, pp. 5-20, 1987.

Kraitchik, M. ``To Guess a Selected Number.'' §3.3 in *Mathematical Recreations.* New York: W. W. Norton,
pp. 58-66, 1942.

© 1996-9

1999-05-25