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We plot $J$ and $K$ $13$ apart. There are a number of options for where to add $L$ and $M$, but working systematically we find that only one option is consistent with $MJ$ being $12$. The final layout looks like

The distance between the points furthest apart is $11+2+12=25$.

*This problem is taken from the UKMT Mathematical Challenges.**View the archive of all weekly problems grouped by curriculum topic*

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