# Sums and Products

On the left side of the applet there is a list of random integers. You select two at a time. (While selecting the second number, hold the Ctrl key down.) When selected, the two integers, say, A and B are replaced with another one computed as (A·B + A + B). The process ends when only one number remains. Controls allow you to select the starting number of integers and maximum possible integer that may be randomly selected into the game.

The final number is not quite random. Can you see whether it's possible to determine it ahead of time?

What if applet does not run? |

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Copyright © 1996-2018 Alexander Bogomolny## Explanation

Assume at a certain stage of the process we have a sequence S of integers, S = {A_{1}, A_{2}, ..., A_{n}}. Let P stand for the product of all integers present, each increased by 1 less 1: P = _{1}+1)(A_{2}+1)...(A_{n}+1) - 1.

It then follows that P is an invariant quantity that does not change in the course of the game. It could be computed at the very beginning. For example, if we are given the sequence

## References

- D. Fomin, S. Genkin, I. Itenberg,
*Mathematical Circles (Russian Experience)*, AMS, 1996

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Copyright © 1996-2018 Alexander Bogomolny