Odd Stones
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.
You may also like
Novemberish
a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.