# Plus or Minus Game

In the simple game below you start with a sequence of numbers. Taking turns you and your computer place either plus or minus sign between any two successive numbers. When all empty spaces have been filled, the arithmetic expression is evaluated. You win if the result is odd. If the result is even, your computer is the winner.

There is a button "+" at the bottom of the applet. Pressing it will toggle "+" and "-". This is to tell you what sign will be inserted upon your click. If the Random checkbox is checked, the applet selects randomly the length of the sequence. Otherwise, it's up to you: try playing short (even very short) sequences first.

What if applet does not run? |

## References

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

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Copyright © 1996-2018 Alexander Bogomolny## Explanation

The parity of the result only depends on the number of odd numbers in the sequence. If the
latter is odd the result will be always odd regardless of the distribution of signs. To see
this pick up a number A. Add and subtract from it another number B: C_{1} = A - B
and C_{2} = A + B. Then, since C_{2} - C_{1} = 2B,
C_{1} and C_{2} are always of the same parity, i.e., either both odd or both even. Thus, the selection
of signs really does not affect the parity of the result. The latter will be odd iff the sequence contains an odd
number of odd members.

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Copyright © 1996-2018 Alexander Bogomolny69654801