Take Ten Sticks

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 ?