## Brain-Teaser 474

**From The Sunday Times, 28th June 1970** [link]

Three friends live at the side entrances to a park bordered by three straight roads. Alan’s entrance is on the shortest road, Colin’s on the longest East to West road, and Brian’s on the road of mid length.

Brian and Colin are directly North and South apart across the park, for half the distance they each are by road from Alan. The distance Brian and Colin are apart, plus the length of the three roads, is three miles.

Each entrance is a number of exact yards by road, from any corner of the park, and Alan is as near to one corner as possible.

How far from that corner is he?

[teaser474]

## Jim Randell 9:06 am

on20 April 2019 Permalink |If B and C’s gates are at a distance of

dapart N-S across the park, and distancesbandcfrom a cornerX, then(b, c, d)form a Pythagorean triple.Once we have a suitable Pythagorean triple we can extend the lines

XBandXCby lengths of2dand bend them (at integer values) so that the bent sides form the third side of the triangle, which meet at A’s gate.If this is possible we have a viable solution. We then need to find which solution places A’s gate closest to a corner of the park.

This Python program finds candidates solutions and choose the one with the smallest distance from A’s gate to a corner. It runs in 108ms.

Run:[ @repl.it ]Solution:The shortest distance A’s gate can be from a corner is 135 yards.The program finds 8 possible solutions:

The last of these has the smallest value for

yorz.So the park looks like this:

However, apparently this was not the intended answer to this puzzle. The puzzle text should have said that the park was a right angled triangle, but this extra condition was omitted when the puzzle was published.

We can remove solution candidates that are not right-angled triangles by uncommenting the check at line 29.

If we do this we find that only one of the solutions remains:

The shortest distance being 275 yards.

In this case the park looks like this:

However, given there is only a single solution it seems odd to have the statement: “Alan is as near to one corner as possible”. It would make more sense to just ask: “How far is Alan from the nearest corner?”.

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