# 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.

### If you are reading this, your browser is not set to run Java applets. Try Firefox and declare the site http://www.cut-the-knot.org as trusted in the Java setup.

 What if applet does not run?

Explanation

## References

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

### Related materialRead more...

• Chessboard
• Changing Colors, an Interactive Activity
• Solitaire on a Circle
• Squares, Circles, and Triangles
• Calendar Magic
• Counting Diagonals in a Convex Polygon
• The Game of Fif
• Nim
• Copyright © 1996-2017 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: C1 = A - B and C2 = A + B. Then, since C2 - C1 = 2B, C1 and C2 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.

Copyright © 1996-2017 Alexander Bogomolny

 61173335

Search by google: