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:

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.

Explanation

References

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

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.