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Programming Enigma Puzzles

12 October 2014

Posted by on **From New Scientist #1375, 15th September 1983** [link]

I’ve been trying to draw the smallest possible circles with a precise number

nof lattice-points on their circumferences. Here are my results forn=4 andn=8. Each has its centre at a point (½, ½), and the radii are ½√2 (about 0.707) and ½√10 (about 1.581).I find the problem trickier when

nis odd. What is the radius of the smallest circle with exactly five lattice-points on its circumference, please? And where will its centre be?

The original article seems to have miscalculated ½√10.

**Note:** I am still waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment. The latest estimate is that I’ll have a connection by the end of October 2014.

[enigma229]

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The program I wrote to find Lattice Circles for

Enigma 136can also be used to solve this problem. It runs in 185ms.Solution:The radius of the smallest circle with exactly 5 lattice points is (25/6)√2 (≈ 5.893). The circle is centred on (1/6, 1/6).The original solution published was that the minimum possible radius is 6.25 (25/4), centred on (1/4, 0), which does indeed give a circle with 5 lattice points, but it is not the smallest possible.

A correction was published with

Enigma 237.