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

Well done Martha in Southend for seeing
this so clearly.
The line before 7 - 1 - 1 - 1 either had 1 or 7 heaps.

If 1 heap (of ten) then the next line would have to be 1 - 9
which isn't what we want.

But 7 heaps will work :

Three of those seven have to be twos so they'd go to ones at the
next move.

That leaves four of the ten sticks for the other four spaces -
so it has to be one in each place.

This means that the arrangement 2 - 2 - 2 - 1 - 1 - 1 - 1 is the
only possible arrangement to go before 7 - 1 - 1 - 1

And could there be something before that 2
- 2 - 2 - 1 - 1 - 1 - 1 arrangement ?
Hamish from New Zealand had similar
reasoning to Fiona (well done Hamish).

He then wondered about a
generalisation.

If 7 - 1 - 1 - 1 can have some thing before it, will any number
of heaps ( arranged as n and the rest ones ) always have something
before it?

Exploring this conjecture, combinations such as 8 - 4 - 1 and 6
- 1 - 1 - 1 - 1 are possible by starting with

2 - 2 - 1 - 1 - 1 - 1 - 1 - 1 and 2 - 2 - 2 - 2 - 1 - 1 - 1 - 1
respectively.

What could come before 4 - 1 - 1 - 1 - 1 -
1 - 1 ?