Part the piles
Age
7 to 11
Challenge level
This is a game for two players.
Begin with a pile of counters. (It might be a good idea to start with, for example, 7 counters.)
The first player must separate the pile into two piles each containing a different number of counters.
Player 2 then splits one of these piles into two unequal piles.
Players take it in turns to separate the piles like this.
The winner is the player who makes it impossible for his or her opponent to divide any of the piles into two unequal piles.
When do you know who will win?
Can you devise a strategy that might help you win?
What happens if you change the number of counters you start with?
This game is a version of the old favourite, Nim. If you search on NRICH for Nim you will find some variations.
Mr Shadbolt, Class 3 teacher at Park Street Primary in Cambridge, sent in the following excellent account of what some pupils did:
We had a go at Part the Piles on Friday.
We tried it with seven multilink cubes. Several of the children in the class -
Maisie, Rebecca, Marco and Matilda - worked out that with seven cubes the person
playing 2nd should always win if they play correctly.
If Player 1 leaves Player 2 with a 6 and 1, then Player 2 should leave Player 1 with a 4, 2 and 1. This forces Player 1 to leave player 2 with 3, 2, 1 and 1.
Player 2 then leaves Player 1 with 2, 2, 1, 1 and 1 which is a losing position.
If Player 1 leaves Player 2 with 5 and 2, then player 2 should leave Player 1 with 4, 2 and 1. The rest of the game then follows on like the one above.
If player 1 leaves Player 2 with 4 and 3 then player splits the smaller pile into a 2 and a 1 which also leaves Player 1 with 4, 2 and 1.
There are only three possible ways for Player 1 to split the pile of seven and all three lead to defeat.
We then tried it with eight and found it much more harder to work out the strategy with the class split on which player has the advantage. Some children think that Player 2 should always win no matter what the number of cubes in the original pile while others thought that Player 1 should win. Some thought there might be a link to whether the start number was odd or even. We might pursue it next week. A fun activity!
Thank you and maybe we will hear more from you again. Well done!
We had a go at Part the Piles on Friday.
We tried it with seven multilink cubes. Several of the children in the class -
Maisie, Rebecca, Marco and Matilda - worked out that with seven cubes the person
playing 2nd should always win if they play correctly.
If Player 1 leaves Player 2 with a 6 and 1, then Player 2 should leave Player 1 with a 4, 2 and 1. This forces Player 1 to leave player 2 with 3, 2, 1 and 1.
Player 2 then leaves Player 1 with 2, 2, 1, 1 and 1 which is a losing position.
If Player 1 leaves Player 2 with 5 and 2, then player 2 should leave Player 1 with 4, 2 and 1. The rest of the game then follows on like the one above.
If player 1 leaves Player 2 with 4 and 3 then player splits the smaller pile into a 2 and a 1 which also leaves Player 1 with 4, 2 and 1.
There are only three possible ways for Player 1 to split the pile of seven and all three lead to defeat.
We then tried it with eight and found it much more harder to work out the strategy with the class split on which player has the advantage. Some children think that Player 2 should always win no matter what the number of cubes in the original pile while others thought that Player 1 should win. Some thought there might be a link to whether the start number was odd or even. We might pursue it next week. A fun activity!
Thank you and maybe we will hear more from you again. Well done!