# Squares and Circles

In the game of Squares and Circles, the players (you and your computer) start with a sequence of shapes: some circles and some squares. On each move a player selects two shapes. These two are replaced with a single one according to the following rule:

Identical shapes are replaced with a square. Different shapes are replaced with a circle.

At the end of the game, when only one shape remains, you are a winner if the remaining shape is a circle. Otherwise, your computer wins.

If the Random box is checked, computer on Reset will randomly select the length of the sequence (from 3 to 20). When the box is unchecked, you may manually set the number of shapes.

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 number of circles is *invariant* under any move. Indeed,
let, before a move, the number of circles be C and the number of squares S. We have to consider three
possibilities:

Selected | After this move | |

circles | squares | |

circle,circle | C-2 | S+1 |

square,square | C | S-1 |

circle,square | C | S-1 |

Therefore, if at the outset the number of circles was odd, you win regardless of the sequence of moves. Otherwise, your computer has the upper hand.

(In a little different guise the problem appears elsewhere.)

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Copyright © 1996-2018 Alexander Bogomolny65812523