## Number Guessing

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 .

The original number is then given by if was Even, or if was Odd. For even, , , , , so . For odd, , , , , so .

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 .
The original number is then given by , since the above steps give .

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.