Start with a number; let’s say 17. Two players take turns to do
subtractions. Each turn a player can subtract 1, 2 or 3 from the number; the
result is then handed to the other player for the next turn; and so on. The player
who reaches zero wins. (That is, the player who has no move loses.)

Observe
that these numbers are important: 0, 4, 8, 12, 16... If we start with a larger
number, the sequence will continue correspondingly. We call these *the key
positions of the game*.

0
is important because whoever reaches 0 wins.

4
is important because if you reach 4, no matter how
your opponent moves, you can reach 0 in your next move. This means whoever
reaches 4 will win (eventually).

8
is important for similar reason. If you reach 8, no
matter how your opponent moves, you can reach 4 in your next move. So whoever
reaches 8 will also win eventually.

These
numbers are multiples of 4 (3+1). n is a key position
if and only if

n mod 4
= 0

Now,
suppose we 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.

What is the winning strategy ?