— ‘The Man Who Mistook His Wife for a Hat’
The original puzzle
We wrote an article for Eureka, the journal of the Archimedeans, the Cambridge University Mathematical Society. It appeared in issue 47, published in February 1987. You can read a PDF version of the article ‘A Song of Six Splatts’: the PDF incorporates scans of the elegant original illustrations prepared by Matthew Richards. Sadly Matthew died in 1993, and the article is reproduced here with the kind permission of his family.
This page summarises the main aspects of the puzzle we invented. Interested readers are urged to consult the original article, which includes much additional material.
The six 3-splatts
The six ‘3-splatts’, as we called them, are shown below: they are given names for convenience.
The original article invited the reader to make a set of 3-splatts from cardboard or expanded polystyrene. Nowadays 3D printing is a viable alternative, and Stewart Russell has kindly prepared suitable files and released them under the Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) licence. You can download the files from Thingiverse here; that page also includes some practical advice on how to get the best results.
With a set of 3-splatts in hand, you can attempt to use them build the following structures.
Note that one of these structures is impossible to make: you can read the original article to find out which!
The forty-four 4-splatts
Four splatts can be joined together in forty-four different ways consistent with their space-filling packing, as illustrated below: names have yet to be assigned to them. The set shown includes both of each of the nine pairs of 4-splatts that are mirror images of one another.
Although the number of pieces in this set is too large to make a satisfying puzzle, it is a pleasing fact that they can be packed into a rectangular cuboid whose sides measure 3 units, 5 units and 8 units. Here the ‘unit’ is the side of the circumcube of the splatt that includes its square faces. Now 3, 5 and 8 are successive terms of the Fibonacci sequence, and so we might call this cuboid a ‘Fibonacci box’. Because the ratio of two consecutive terms in the Fibonacci sequence approaches the ‘golden ratio’, four of the faces of the box approximate ‘golden rectangles’, which accounts for its handsome appearance. The box as a whole approximates the ‘golden cuboid’, apparently the favoured sitting-room shape of audiophiles.
The illustration below shows this cuboid, the colours hinting at one solution to the packing problem.
How many n-splatts are there?
The number of distinct n-splatts is given in the table below. For n≥4 there are two different counts, depending on whether two splatts that are mirror images of one another are counted as distinct. Entries for n≤8 are confirmed by The On-Line Encyclopedia of Integer Sequences (sequences A038180 and A038181); I would welcome independent confirmation (or contestation) of the later entries as well as any results that extend the table.
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