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 
from an audience member.
1. Ask the person to compute 
(i.e., three times the secret number 
) and announce if the result is even
or odd.
2. If you were told that 
is even, ask the person to
compute the number 
which is half of 
.
If you were told that 
is odd, ask the person to compute the number 
which is half of 
.
3. Ask the person to compute 
.
4. Ask the person to divide 
by 9 and to reveal the quotient 
, discarding any remainder.
The original number 
is then given by 
if 
was even, or 
if 
was odd. For 
even, 
, 
, 
, 
, so 
. For 
odd, 
, 
, 
, 
, so 
.
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 
.
The original number is then given by 
, since the above steps give 
.
