*Subtraction
Game with two numbers*

Start with two numbers; let’s say 9 and 14. Each turn a player can
subtract 1, 2 or 3 from one of the numbers. The one who makes the last move
wins.

We
find the winning strategy by finding the key positions. What are the key
positions for a two numbers subtraction game ?

If
one of the numbers reaches zero, the game would become a one number subtraction
game. So the position (0, n) is a key position if and only if

n mod 4 = 0

What
are the other key positions?

(Follow
the Power Point presentation to find out more
… You can use OpenOffice to read it.

It
looks better in presentation format. If you don’t have the presentation programs,
try pdf file.)

## Advanced Topic

Read on if you know what does *n mod t = m
mod t * mean.

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