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## 'Odd Stones' printed from http://nrich.maths.org/

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.