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### Number and algebra

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### Advanced mathematics

# Take Ten Sticks

#### Take ten sticks and put them into heaps any way you like.

#### Now try other starting distributions for the ten sticks.

#### If you're getting intrigued by the patterns you'll definitely want to generalise your result.

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Age 11 to 16

Challenge Level

One possible distribution of the sticks is 4 - 1 - 5, but there are lots of other arrangements possible.

Next make one new heap using a stick from each of the heaps you have already.

Our example now becomes 3 - 3 - 4 (notice how the heap with just one stick vanishes).

Then keep repeating that process : one from each heap to make the new heap.

So the next thing we get is 3 - 2 - 2 - 3, followed by 4 - 2 - 1 - 1 - 2 .Continue repeating this until you see the distribution settle in some way.

You can of course begin with more, or less, than three heaps.

Could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?

That's the main question, but you may like to pose yourself other questions about this situation.

Let us know what you try and what you find out.

Mathematicians like to notice patterns and then try to explain them, can you explain any of the things you noticed?

Eleven sticks, twelve, any number, explaining as much as you can about what you notice.

Have fun!

Printable NRICH Roadshow resource.