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A Stanley puzzle

The subject of this post is yet another ebay purchase, this time made purely on the basis of a name on a box. Described as geometrical models for teaching, the lot consisted of two sets of “Invicta” Plastics Technical Drawing Models and a single cardboard box bearing the 286 High Holborn address of W.F. Stanley & Co. Ltd., the latter being the item that had caught my attention.

The purpose of the Invicta models was helpfully outlined on the lids of their boxes, being intended to accompany the series of three Technical Drawing Course books by M. Hirst.

Invicta Plastics Technical Drawing Models in original box

Made of high impact polystyrene, they probably dated from the 1960s or later and had clearly been well used in an educational setting.

The Stanley box was similar in construction to the ones used for their loose compasses and half sets, made of a stout brown card with reinforced metal corners.

W.F. Stanley Stomachion set c.1926 original box

It was earlier than the Invicta set, dateable to between 1924 and 1931 due to the combination of High Holborn address, early telephone number and telegram code, along with a cancelled three penny postage Stamp from the reign of George V (produced until around 1924, but still valid for postage indefinitely thereafter).

W.F. Stanley Stomachion set c.1926 George V three pence stamp from original box

Unfortunately the date of its cancellation mark was too indistinct to read, although the words around the perimeter could be made out as “SUNDERLAND Co. DURHAM”.

Inside was a jumble of pieces made of boxwood and what appeared to be dense white celluloid or bone, in a range of triangular and irregular polygonal shapes.

W.F. Stanley Stomachion set c.1926 unassembled in original box

Some of the pieces were marked “STANLEY LONDON” and “MADE IN ENGLAND”, in keeping with the date of the box, but strangely these words were cut up almost like a jigsaw puzzle. Unlike the Invicta sets, there was no indication as to their intended use.

My first attempts to find out more about this mystery box succeeded only in tracking down a recent auction lot from which all three sets had been split off for resale (fortunately I had paid very little for my portion).

Auction lot of assorted geometric models

The original lot entitled Solid geometry models etc. also contained two sets of Helix geometrical solids in wood and a set of printed card Descriptive Geometry Models by Thomas Jones, all of which had been sold separately on ebay at a reasonable profit. Sadly the auction listing did not furnish any additional details, noting only that the Stanley box contained “geometric shapes in boxwood with another in celluloid, c.1920”.

Having hit a dead end, I decided to take an empirical approach to the problem. With the pieces all spread out on a large tray, I attempted to form them into some kind of pattern. It soon became apparent that there were three sets of tiles marked with Stanley’s name, two on the white pieces and one on the boxwood. Moreover, the words occupied the same positions on similar tiles, which suggested that I was dealing with three distinct groups of pieces, one boxwood and two white.

By dividing up the pieces into three groups, each containing similar shapes, I had some early success in arranging the boxwood set into a geometric form consisting of a square and triangle with the Stanley inscription intact, but somehow this did not feel entirely satisfactory.

W.F. Stanley Stomachion set c.1926 alternative arrangement

At this point I made up my mind to cheat by trying to match the boxwood pieces according to the grain of the wood, which was distinct enough to make this possible.

To my astonishment it worked, and I finally had a completed puzzle in the shape of a rectangle of 2×1 proportion.

W.F. Stanley Stomachion set c.1926 canonical arrangement of Richard Dixon Oldham

Following the same arrangement, I was able to assemble the two white sets, thankfully with no pieces missing. So the box appeared to contain three separate sets of tiles, each forming the same rectangle. They even fitted nicely into the box in their completed state!

W.F. Stanley Stomachion set c.1926 assembled in original box

In addition, because each square was divided diagonally, the four quarters could be rearranged to form a larger square (in a diamond orientation if Stanley’s signature be taken into account).

W.F. Stanley Stomachion set c.1926 three squares

Even so, I still had no idea what the purpose or meaning of the tiles were, so I resumed my search for connections online, hoping to find some inspiration.

The first thing that came to mind was a plastic puzzle that I remembered having as a child, also consisting of triangles and polygons that could be arranged into a square. From these vague memories, I eventually tracked it down to the Euclid puzzle game by Kohner Bros. Although named after an ancient Greek mathematician, this is in fact a classic tangram puzzle. The tangram apparently came from China around the late 18th century and soon became popular in the West. Its name seems to come from the obsolete English word trangam (misspelt in Johnson’s 1755 dictionary as “trangram”) meaning a trinket, puzzle or odd gadget. By 1806 the word appears in its present day form in J. Walker’s Rhyming Dictionary.

The standard tangram had seven pieces, whereas Stanley’s rectangles as I had assembled them have fourteen. At one level this makes sense, because Stanley’s consist of two squares, compared to the tangram’s one. However, the Stanley pieces are much less regular than those of the tangram and certainly can’t be arranged in the same way. Fortunately the plastic tangram toy incongruously named after Euclid had reminded me of something else: Euclid’s 47th Problem.

This is the name given to Euclid’s proof of Pythagoras’ earlier well-known theorem that for a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides. It formed the centrepiece of the first book of Euclid’s Elements written around 300 BC, in the guise of the elegant windmill proof (which happily gives me another opportunity to link to the beautiful version of Byrne’s Euclid)

Why this was relevant to the Stanley puzzle is because of another childhood memory of having seen geometric solutions that involved cutting up the two smaller adjoining squares into shapes which could be rearranged to form the large square. These so-called tangram proofs are thought to have been first devised in the third century by Chinese mathematician Liu Hui.

Tangram-style proofs of Euclid's 47th Problem

There are several variants of the tangram proof, including the one ascribed to Liu Hui and a version using the seven standard tangram pieces (below), but however I tried, Stanley’s pieces resisted being arranged into Euclidean shapes.

Solution of Euclid's 47th Problem using standard tangram pieces

A similar puzzle called the Haberdasher’s Problem involves dissecting an equilateral triangle so it can be rearranged as a square, but this again is too simple for the Stanley set.

Habedasher's problem showing both end states with unfolding method below

If it wasn’t any of these, perhaps it was a variation of the tangram game, which is where I turned next. One of the earliest sources to use the name tangram in relation to the Chinese puzzle seems to have been the short 1848 book Geometric Puzzle for the Young by Thomas Hill, who was later to become President of Harvard University. Hill’s puzzle was a simplification of the tangram for educational use, being composed only of isosceles right angle triangles to be arranged into the patterns suggested in the pages of his book. While these are clearly unrelated to the more complex Stanley pieces, the book’s introduction makes a tantalising reference:

“The value of these puzzles, in cultivating a geometrical taste and ability, has been acknowledged since the time of Archimedes, who is said to have invented a similar play”

This mention of a geometric game by Archimedes gave me a new lead to follow. Known variously as Ostomachion, Stomachion or Loculus Archemidius, the “game” was devised as a fourteen piece dissection puzzle that could form a square, as well as being used for creative shape-making exercises. The name Ostomachion is said to derive from the ancient Greek words for “bone fight”, possibly due to the pieces being originally made of bone, while the Latin Loculus Archemedius means Archimedes’ box, presumably because the pieces could be arranged in a box shape (although there is some suggestion that it may refer to the geometric design of the Roman loculus or satchel).

The puzzle was derived from a treatise by Archimedes also called Ostomachion, or Stomachion, of which only fragments survive: one is an Arabic translation and the other a Byzantine Greek manuscript known as the Archimedes Palimpsest (a palimpsest is a text that has been erased so that a new text could be written over the top, a common practice in the middle ages). The Arabic version had been translated into German in 1899 by Suter, who had also reconstructed a diagram of the puzzle.

Stomachion diagram Fig. 1 from R.D. Oldham's article in Nature, March 6, 1926

Now it felt like I was finally getting somewhere. Archimedes’ puzzle had the correct number of pieces, and the shapes looked similar. Yet the ostomachion after Suter was strangely stretched compared to Stanley’s – it still didn’t quite stack up. The last line of the Wikipedia page gave me another clue, namely that “surviving images of the puzzle show it in a rectangle, not a square”. Where were these images from? The Archimedes Palimpsest was a possibility, so that was my next destination.

The Archimedes Palimpsest was, as its name suggests, a Greek copy of various works by Archimedes that had survived the sack of Constantinople in 1204 by being smuggled to a remote Greek monastery in Palestine, where it was soon overwritten with religious texts. Here it remained unknown until 1899 when it was catalogued and subsequently recognised by the Danish historian Johan Ludvig Heiberg as a lost work of Archimedes. Heiberg was allowed to photograph the pages in 1906, which he published to international acclaim in 1915. However, the palimpsest went missing in 1922 and was not seen again until 1998 when it was put up for auction by Christie’s (the story of which is an intrigue in its own right). The manuscript sold for more than two million dollars, double its estimate, the purchaser said to be Amazon founder Jeff Bezos. The entire book has now been digitised and both images and transcription are available online (Google Books version here).

Even so, this does not explain how Stanley’s version came to resemble the rectangular palimpsest image, as the 1915 publication by Heiberg uses the same square format for the full 14-piece dissection as Suter’s derived from the Arabic, above. The answer can be found in a letter submitted to the journal Nature in 1926 by renowned British geologist Richard Dixon Oldham.

Stomachion article by R.D. Oldham in Nature, March 6, 1926

Here was the very layout used by Stanley, based on a double square, as reconstructed by Oldham from his reading of Heiberg’s palimpsest text (incidentally, it shows that Stanley’s name is actually printed on the back of the puzzle, not the front).

Stomachion diagram Fig. 2 from R.D. Oldham's article in Nature, March 6, 1926

As well as departing from the clearly erroneous square format, Oldham’s reconstruction differs in two significant details from the earlier versions – shown as dotted lines in his illustration – thus cementing its connection with the Stanley set. Oldham even revisited the origins of its Greek name, concluding that the correct form is in fact “Stomachion” (meaning “the thing that drives one wild”) as used in the palimpsest, while “Ostomachion” is a later corruption. Aside from its obvious mathematical interest, Oldham also elaborated on the ancient concept of stomachion as a kind of creative shape-making game, of which he included several examples.

Stomachion examples Fig. 3 from R.D. Oldham's article in Nature, March 6, 1926

In the spirit of the game, I have made my own festive effort!

W.F. Stanley Stomachion set c.1926 reindeer arrangement

As this new evidence suggested that my Stanley set was unlikely to predate 1926, a search of publications from that year led to the discovery that the stomachion game had become something of a craze after the appearance of Oldham’s piece in Nature, not only in Britain, but also in the USA. An article in the Washington Post of Sunday June 27, 1926, introduced the newly-discovered puzzle in the column “Fashions of Capital Women”.

Stomachion article and extracts from Fashions of Capital Women in The Washington Post, June 27, 1926

It opens with a short account of stomachion’s rapid ascent:

“With the departure of “mah jong” as the ultra-fashionable game and the arrival of “stomachion,” fostered by the Prince of Wales and his London set, we may expect to see the mah jong styles replaced by those of the new pastime. For it is certainly true that each new game fad brings in an accompanying clothes vogue.
Stomachion, which is really 2,000 years and therefore a revival, bids fair to be the “game” of the summer, at least among those interested enough to obtain a set of its pieces and to learn how to play it.”

The Prince of Wales was of course the later Edward VIII of abdication notoriety, whose infatuation with all things American was well known. The article then goes on to describe a familiar-sounding object, the origin and details of which are consistent with Stanley’s product:

“The ancient game dovetails in most admirably with the modernist ideas of color and design. The game sets already seen here in several houses where the hostess has returned from London this spring are composed of rectangles of ivory or ebony pieces, cut out by a jig saw into 14 triangular and polygonal pieces, according to a scheme devised by the original inventor of the game, Archimedes.”

Oldham’s involvement is also acknowledged, under the heading “Research Revives Game”:

“So scientific is this game that R.D. Oldham, geologist of the Royal Society in London, constructed the first modern set after patient research through newly-discovered manuscripts in Greek, Latin and Arabic. Mr. Oldham, who is a scientist, first last and all the time, has become suddenly much sought after by the smartest members of London society, for not only is he an oracle on the intricacies of the game as played by the Greeks, but he also holds in his hand the key to the making of the mystic pieces which can not be procured at the present time in any other way. His first set made, after careful research, was the first one made in 1,500 years.”

It is unclear whether the expression “can not be procured at the present time in any other way” meant that stomachion sets were not yet in production, or else that only Oldham’s official sets were available in London, but the preceding reference to ivory and ebony sets being brought back by travellers suggests the latter.

As with most fads, stomachion’s time in the limelight was brief, and by 1930 it had been relegated to the Junior Post section of the same newspaper, this time as a children’s game that could be cut out from cardboard at home.

Stomachion article from The Junior Post Section of The Washington Post, June 29, 1930

It is perhaps appropriate that, having faded from fashionable circles, stomachion should return to its original purpose as an educational game.

The last question that remains is exactly how Oldham’s stomachion came to be made by Stanley. I believe the answer may lie in a professional connection between the two men, specifically through the Geological Society, of which William Ford Stanley had also been an active member. Oldham is best known for the discovery of the earth’s core, based on his study Report on the Great Earthquake of 12th June 1897, made while working for the Geological Survey of India.

Stanley was elected as a fellow of the Geological Society of London in January 1884, and subsequently went on several expeditions to Ardennes, Egypt, Palestine and Switzerland in connection with the society’s activities. Oldham meanwhile was a lifelong professional geologist, awarded the society’s Lyell Medal in 1908 (a year before Stanley’s death) and later serving as president from 1920 to 1922.

It is therefore inconceivable that the two did not know each other, and certainly Oldham will have known Stanley’s surveying instruments through his work with the Geological Survey of India. Although by 1926 Stanley himself had long since died, it is probable that when Oldham looked to have his stomachion sets made his thoughts turned to the firm, well known for the quality of its work in ivory, ebony and boxwood.

W.F. Stanley Stomachion set c.1926 square arrangement with incomplete puzzle

Whether the connection was an ongoing personal one – perhaps via Stanley’s successor H.T. Tallack who particularly specialised in surveying instruments – or simply a result of Oldham’s longstanding professional links with the firm may ultimately be difficult to establish.

One thing however is certain: Oldham’s definition of stomachion as “the thing that drives one wild”. After writing this post I can vouch for that!

This Post Has 2 Comments

  1. Alan Williams

    Thanks Erick, that is a really interesting resource you link to! At least some of the Richter puzzles are based on the classic tangram, which the more I look at it seems to be a rationalised version of the Stomachion. I wonder if the puzzle could have made its way to China via the Arabic mathematical treatises, before eventually making its way back to Europe and America in its modified form.

    Another thing from your link that appeals to me is the full set of tangram letters and numbers found in the Anchor puzzle’s instruction booklet, each figure using all of the available pieces. As these are helpfully all printed to the same scale, I have assembled them alphabetically as a single image that I will post to the blog.

    It’s funny that the German name for the puzzle was “Kopfzerbrecher”, meaning headache, being used to promote Dr Richter’s other line of business in pain-relief medication (I assume his pharmaceutical business preceded the puzzles). Thanks again for your input!

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