Take Ten Sticks
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!
Printable NRICH Roadshow resource.To get 7 - 1 - 1 - 1 how many heaps could there have been in the line before?
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 - 1And 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 ?
This problem is good to encourage systematic thinking - working from a simple start: two sticks, three, four . . .
Encourage questions like "what could come before . . . . . and what before that" ?
Working backwards from the goal is a mathematical thinking skill.
Especially encourage explanation - accounting for pattern.
There is lots of interest in this simple mechanical process - look out for the triangle numbers!
Above all, if possible, let each insight prompt fresh questions to consider, each new question will illuminate more of the structure.