## November 02, 2008

### The 144 Game

My son is learning factors, multiples, and primes in school. That means we can play the 144 game. (You could actually use any number.) It also means we have a new puzzle for you.

Here is how you play.

Two players take turns naming positive integers. The only rules are that:

1. Every number played must be a factor of 144.
2. You may not play a factor of any previously played number.
3. The player who is forced to say "144" loses.

For example, you could play:

Player 1: 1
Player 2: 24
Player 1: 18 (2, 3, 4, 6, 8 or 12 not allowed, but 18 is okay.)
Player 2: 72 (generally not a good move)
Player 1: 48 (checkmate)
Player 2: 144 (oops, player 2 loses)

Player 1 has an advantage. Why? Can you figure out the right first move?

Posted by David at November 2, 2008 08:24 AM

I, as player 1, open with 12. Valid moves remaining are {18,24,48,72}. This is a nice small set, and we can reason about each case in turn.
So, if you play 18, then I follow up with 24 and
you have to go to 144.
On the other hand, if you try to curtail my options by picking 24, I play 18 and you lose.
If you play 72, I play 48, you pick one of the two remaining, I pick the other, and you're forced to 144.
If you play 48, I play 72, and then play proceeds as per the previous case.

In practice, then, after player 1 plays 12, they keep the following two pairs in mind: {18,24} and {48, 72}. If their opponent plays an element of a pair, they play the other element.

I'd like to try to find an elegant proof for why that strategy works, but at the moment I'm a bit too sleepy.

Posted by: Iain at November 2, 2008 09:11 AM

After 12, actually there are more valid moves - 8, 9, 16, 36. Can you find a way to visualize the game?

Player One has an advantage in games for not just 144, but any number larger than 1. There is a nonconstructive argument why this is true...

Posted by: David at November 2, 2008 05:05 PM