Copyright © University of Cambridge. All rights reserved.

## 'Take Ten Sticks' printed from http://nrich.maths.org/

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

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.

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

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?

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

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

Have fun!