**From New Scientist #2920, 8th June 2013** [link]

I have drawn a circle, marked five points around its circumference, and joined each to the next by a straight line in order to make a pentagon. It turns out that the centre of the circle is outside this pentagon. I have then measured, in degrees, the five interior angles of the pentagon. The five numbers are all different and all but the smallest are perfect squares.

What is the smallest angle and what are the angles on either side of the smallest one?

[enigma1752]

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This Python program runs in 44ms. I’ll provide a diagram to explain the solution later.

Solution:The smallest angle is 42°. The angles on either side of it are 64° and 121°.Here’s the diagram that explains my approach:

ABCDE is the pentagon inscribed in the circle centred on O.

Radii from A, B, C, D, E to O form 4 isosceles triangles, so the angles at the vertices on the circumference of the circle for each triangle are the same – a, b, c, d, and on the triangle AEO the angles at the vertices on the circumference are both f.

So we find the angles such that the internal angles of ABCDE at A, B, C, D are perfect squares – these angles are: a – f, a + b, b + c, c + d. The angle at E (the smallest angle) is d – f.

Also the internal angles of the pentagon sum to 540°, so:

hence:

And since (a + b) and (c + d) are integers, then so is f.

It’s more efficient to choose the squares and derive f. Although it only makes a minor difference to the runtime. This program’s overall run time is 40ms, but it examines many fewer options.

This uses a different approach

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My approach is totally different, I guess, before coding the enigma, I calculated the possible maximum 3 angles, and after having determined the limits of those values, and checking the center angle of the circle, I did write the code, I think this is enough fast