Here is a simple trick based on group theory. Ask a person to choose four numbers from 1 to 9 and write them in a row on a piece of paper. Pause for a moment and then write a number on a piece of paper without letting the other person see what it is. Turn the paper over and place it on the table.
Now ask the person to choose two of the numbers from the list and put a line though them. Ask the person to compute a*b + a + b and put it in the list to replace the two chosen numbers.
Continue to do this until there is only one remaining number. Turn over the paper and show that the numbers match.
The simplest way of explaining this is to show that a * b + a + b is isomorphic to multiplication using the transform T(x) = x + 1. (a*b + a + b) + 1 = (a + 1)(b + 1). If we denote the operation a * b + a + b as a & b, this means that a & b is commuative and associative, just as multiplication is. For any list of numbers ai, the final number can be computed as the (a1 + 1)(a2 + 1)...(an + 1) - 1.
