# Sums and Products

This trick is a little more complex than Splitting Piles. It may easily lead to larger numbers. For a demonstration you will probably need a board or a piece of paper to write on. Write (or make somebody write for you) several numbers.

Select a spectator from the audience. Explain to the person and the audience what you expect them to do. Leave the stage while they are following your bid. Ask them to call you when finished. As the result of their activity, they will produce a number. Upon your stepping back on the stage you declare that you know what the number has been generated, and eventually reveal the number.

This is what the fellow has to do in your absence. He (or she) will proceed in several steps. On every step, the fellow will select any two of the numbers and replace them with another one according to the following rule: numbers A and B are replaced with (A·B + A + B). The process stops when only one number remains. This is the number you are going to "guess."

For example, let start with the sequence 1,2,3,4. (Although the numbers you write must not necessarily be all different. Let's agree to always replace the two leftmost numbers.

Thus, first you replace 1 and 2 with (1·2 + 1 + 2) which is 5. Now there are only three integers: 5, 3, 4.

Next you replace 5 and 3 with (5·3 + 5 + 3) which is 23. You are left with 23 and 4. These are replaced with (23·4 + 23 + 4) which is 119. This is the number you will "guess" upon returning to the stage.

There is an additional page with a Java applet that lets you experiment with this trick. Try it before reading the explanation of how it works.