## Teaser 2917: Polygoland

**From The Sunday Times, 19th August 2018** [link]

We’ve all heard of Heligoland but George and Martha are on holiday in Polygoland. This is an island and when it was first inhabited, the authorities created as many national parks as possible subject to the restriction that each such park will be a regular polygon with the angle between each side being an integral number of degrees, and all the parks have different numbers of sides. Walking along the border between two such parks, Martha commented that the angle (A) of one of them was an exact multiple of the number of sides (S) in the other. George mentioned that the multiple was equal to the number of parks on the island.

What are A and S?

[teaser2917]

## Jim Randell 3:40 pm

on1 May 2019 Permalink |For a regular

n-gon, the internal angles are(180 – 360 / n)degrees.So we can determine the number of possible

n-gons with integer anglestfrom the number of divisors of 360 (greater than 2).There are 22 possible polygons, so

t = 22So we need to find a solution to the equation:

where

nis a divisor of 360 (greater than 2).There are only a few values of

mto consider and only one of them gives a viable value forn.Run:[ @repl.it ]Solution:A = 176°, S = 8.The solution to the equation is

m = 8, which gives a value ofn = 90.The angle 176° is 4° off a straight line, so if we walk along the perimeter each side turns us by 4°. After 90 sides we have turned through 360°.

176 = 22 × 8, and 8 is a divisor of 360, so there will be an octagonal park (with interior angles of 180° – 45° = 135°).

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