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 queries that can be used to determine an unknown number n from an audience member.

1. Ask the person to compute n^'=3n (i.e., three times the secret number n) and announce if the result is even or odd.

2. If you were told that n^' is even, ask the person to compute the number n^('') which is half of n^'. If you were told that n^' is odd, ask the person to compute the number n^('') which is half of n^'+1.

3. Ask the person to compute n^(''')=3n^('').

4. Ask the person to divide n^(''') by 9 and to reveal the quotient k, discarding any remainder.

The original number n is then given by 2k if n^' was even, or 2k+1 if n^' was odd. For n=2m even, n^'=6m, n^('')=3m, n^(''')=9m, k=m, so 2k=2m=n. For n=2m+1 odd, n^'=6m+3, n^('')=3m+2, n^(''')=9m+6, k=m, so 2k+1=2m+1=n.

Another method asks:

1. Multiply the number n 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 n^'.

The original number is then given by n=(n^'-165)/100, since the above steps give n^'=5(4(5n+6)+9)=100n+165.

See also

Number Picking

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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.Chandrasekaran, K. R. "Think of a Number from 1 to 27.", S. and Flannery, D. In Code: A Mathematical Journey. London: Profile Books, p. 66, 2000.Kraitchik, M. "To Guess a Selected Number." §3.3 in Mathematical Recreations. New York: W. W. Norton, pp. 58-66, 1942.

Referenced on Wolfram|Alpha

Number Guessing

Cite this as:

Weisstein, Eric W. "Number Guessing." From MathWorld--A Wolfram Web Resource.

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