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Turn $2$ - $8$ - $17$ into $4$ - $7$ - $16$
then $4$ - $7$ - $16$ into $3$ - $9$ - $15$
This example isn't the same thing but might give you a clue about the kind of thinking to try.
In a $4$ circle problem and using $26$ stones the distribution $1$ - $4$ - $7$ - $14$ cannot be turned into $3$ - $5$ - $7$ - $11$
To understand why notice that in the first there are two odd and two even numbers while in the second the numbers are all odd.
On a "move" one value goes up by $3$ and the others go down be one.
What will happen to odd and to even numbers?
The Odd Stones problem isn't about odd or even numbers but a similar kind of thinking could be useful.