Games of Strategy

The Game of Nim

This is a very old game, there are lots of different versions of Nim.

Game 1:
Play against a partner.
Use 15 counters (or you could use coins or matches).
Take it in turns to pick up 1, 2 or 3 counters at a time.
The winner will pick up the last counter.

Can you work out a winning strategy?

You could also vary this game by using a different number of counters to start with, or by picking up different numbers.  You could also try playing that you lose if you pick up the last counter.

What effects do these changes have on your winning strategy?