# Enigmatic Code

Programming Enigma Puzzles

## Enigma 229: Five-point circle

From New Scientist #1375, 15th September 1983 [link]

I’ve been trying to draw the smallest possible circles with a precise number n of lattice-points on their circumferences. Here are my results for n=4 and n=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 n is 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]

### One response to “Enigma 229: Five-point circle”

1. Jim Randell 12 October 2014 at 8:28 am

The program I wrote to find Lattice Circles for Enigma 136 can also be used to solve this problem. It runs in 185ms.

```% python lattice-circles.py 5
[using gmpy2.mpq]
[checking radius < 1] [nmin=3] [4 points in 1/1 sector] ...
-> n=4 r=0.707106781187 (r^2=1/2) (1/2, 1/2)
[checking radius < 2] [nmin=3] [16 points in 1/1 sector] ...
-> n=3 r=1.66666666667 (r^2=25/9) (1/3, 0)
-> n=8 r=1.58113883008 (r^2=5/2) (1/2, 1/2)
-> n=3 r=1.17851130198 (r^2=25/18) (1/6, 1/6)
[checking radius < 3] [nmin=5] [20 points in 1/2 sector] ...
-> n=6 r=2.5 (r^2=25/4) (1/2, 0)
[checking radius < 4] [nmin=5] [26 points in 1/2 sector] ...
-> n=12 r=3.53553390593 (r^2=25/2) (1/2, 1/2)
[checking radius < 5] [nmin=5] [32 points in 1/2 sector] ...
[checking radius < 6] [nmin=5] [44 points in 1/2 sector] ...
-> n=5 r=5.89255650989 (r^2=625/18) (1/6, 1/6)
-> n=16 r=5.7008771255 (r^2=65/2) (1/2, 1/2)
n=5 r=5.89255650989 (r^2=625/18) (1/6, 1/6)
```

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.

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