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Programming Enigma Puzzles

21 January 2015

Posted by on **From New Scientist #1399, 1st March 1984** [link]

Each of the black dots in the picture is one of the elements — 3 speakers and 22 suppressors — in Uncle Pinkle’s Triphonic Asymmetric Garden Audio-system. The whole square array measures 40 × 40 metres. Uncle Pinkle sits at

C, precisely the same distanceDfrom each speaker, but not at distanceDfrom any suppressor; andCmust not be directly between any two system-elements, so it isn’t on any of the vertical, horizontal or diagonal lines in the picture. Within these restrictions, Uncle Pinkle has arranged the system-elements and placed his seat so thatDis as small as possible.How far

isthe distanceD? (An answer to the nearest centimetre will do).

[enigma252]

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This is lattice circle problem, similar to

Enigma 136andEnigma 229(also by Stephen Ainley).I adapted the lattice circle solver I wrote for

Enigma 136to solve this.This Python program runs in 555ms.

Solution:D is 18.21m (or (25/7)√26).This diagram shows one of the 32 possible positions for C. The position C is marked with a large cross, and the three equidistant speakers are marked with circles. The 31 alternative positions for C are marked with smaller crosses.

The example position for C is at (95/7, 85/7) (= 10 × (19/14, 17/14)).

I don’t normally post my solution if it is too similar to your own and this one certainly is in terms of approach. But there are a number of practical differences so I decided to post in this case.

As you found in the related lattice based teasers, the GMPY2 implementation of rational fractions is much faster than that in Python (as posted it uses the latter).

The numbers would have been neater if the grid spacing in Uncle’s array had been, say, 28 ft.

Then the coordinates of C (measured from the corner speaker) would be (34, 38) or (38, 34); and the distance D would be sqrt(2600) which is very close to 51.

I wonder what his “suppressors” consist of. Think I’ll stick to indoor stereo myself!