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Odd Stones

Age 14 to 16 Challenge Level:

$27$ stones are distributed between $3$ circles


On a "move" a stone is removed from two of the circles and placed in the third circle.

So, in the illustration, if a stone is removed from the $4$ and the $10$ circles and added to the $13$ circle, the new distribution would be $3$ - $9$ - $15$

Check you can turn $2$ - $8$ - $17$ into $3$ - $9$ - $15$ in two "moves"

Here are five of the ways that $27$ stones could be distributed between the three circles :

$6$ - $9$ - $12$

$3$ - $9$ - $15$

$4$ - $10$ - $13$

$4$ - $9$ - $14$

$2$ - $8$ - $17$

There is always some sequence of "moves" that will turn each distribution into any of the others - apart from one.

Identify the distribution that does not belong with the other four.

Can you be certain that this is actually impossible rather than just hard and so far unsuccessful?