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I originally solved this when it was published in the magazine using Perl that uses code to check two possible cases (determined by analysis). Here’s a similar approach in Python, it runs in 52ms.

Solution:Jay and Kay were 7km apart.This diagram shows where the derivation of the equations used in the program.

By considering the similar triangles JIJ’ and ITJ (or ITK) it follows that:

Hence a and t are integer factors of 16 and a < t, so: (a, t) is (1, 16) or (2, 8).

And we can eliminate one of these possibilities to give a unique solution by further analysis.

Considering the quadrilateral IJ’TK we see that:

Hence IJ’TK is a cyclic quadrilateral, so by applying Ptolomy’s Theorem to it we get:

so:

But x is an integer so t divides 4a, leaving (a, t) = (2, 8) as the only possible solution.

Hence x = 7.