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Fifteen game
By Rachelle Najman (T4009) on Monday, April 16, 2001 - 06:21 am:

Nine counters with the numbers 1 to 9 are placed on the table. Two players alternatively take one counter from the table. The winner is the first player to obtain, among his counters, three with the sum exactly 15.

- what is the best move to take?
- does the sum 15 in the context of the numbers 1 to 9 ring a bell?
- what sets of numbers sum to 15?
- how many times does each number occur in these sets?
- can you arrange the sets so that their overlap
is displayed?

[This game can be found here. - The Editor]

By Anonymous on Monday, April 16, 2001 - 09:54 am:

Try arranging the numbers in a magic square...

By Rachelle Najman (T4009) on Monday, April 16, 2001 - 06:38 pm:

huh?

By Olof Sisask (P3033) on Monday, April 16, 2001 - 07:07 pm:

I think Anon means:
Try arranging the numbers in a 3x3 square so that the sum of the numbers in each row and each column is 15 (a square such is this is called a magic square).

By The Editor:

There are some articles about magic squares on the NRICH site.

If you arrange the numbers 1 to 9 in a magic square, and then play the fifteen game, you may notice that the game is exactly equivalent to a very well-known pencil-and-paper game. Any strategy you have for one of the two games will easily transfer to the other. In particular, many people know a strategy that means they will never lose at the pencil-and-paper game, and this can be turned into a strategy for never losing the fifteen game.

This equivalence between two apparently different situations is very important in advanced mathematics: it means that if you can prove a result in one situation, you can prove it in the other as well.

See here for some other games to analyse.