Seven Regions

Place the digits $1$ to $7$, one in each region, so that the circles all have the same total.

Can you also show that:

- you cannot have a circle total of $16$ with $4$ in the centre?
- you cannot have circle totals greater than $19$ or less than $13$?
- you cannot have anything other than $1$ in the centre for a circle total of $13$?

Five Rings

These five rings create nine regions, labelled $a$ to $i$ above. Using each of the digits $1$ to $9$ exactly once, can you place one number in each region so that

**the sum of the numbers within each ring is the same?**
Can you find more than one solution?

Show that for any solution the sum of the numbers in the overlaps ($b$, $d$, $f$ and $h$) must be a multiple of $5$.

Using this, can you find a lower and an upper bound for the possible ring totals?

Is there a solution for every ring total between the lower and upper bound?

If not, can you prove that no such solution exists?

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*If you enjoyed this problem, you may also like to take a look at Magic Letters.*

*With thanks to Don Steward, whose ideas formed the basis of this problem.*