Good start from Hamish in New Zealand

2 - 8 - 17 goes to 4 - 7 - 16

and from there to 3 - 9 - 15

Ruth from Manchester High School then looked for the odd one out

4 - 9 - 14 is the impossible arrangement.

Consider the numbers in each circle in modulo 3.

Modulo 3 means the remainder amount when you divide a number by three.

In the first arrangement (6 - 9 - 12) the modulo 3 value of each pile is 0 .

On each move you take 1 from 2 of the piles and add 2 to the third so the numbers which were all 0 in modulo 3 now all become 2 in modulo 3, and after that 1 in modulo 3, then finally 0 again.

After that the cycle just repeats over and over again.

For four of the arrangements the initial numbers are all equal in modulo 3 and whatever you choose as the next move they will stay equal in modulo 3.

But 4 - 9 -14 is 1 - 0 - 2 in modulo 3 and so cannot turn into any of the other four arrangements or be reached from them.

Thanks Ruth. That way of looking at numbers using their modulo value seems like a powerful perspective.