The stomachion is perhaps the world's oldest puzzle. It was described by the mathematician Archimedes of Syracuse (Αρχιμήδης ο Συρακούσιος) in the 3rd century BC, but his words have survived only fragmentarily. It consists of 14 pieces drawn from a square. Throughout the ages, people have amused themselves with the design of new shapes with these pieces. Animals, buildings, stars and other interesting shapes can be made with them, not unlike the Chinese tangram. Although it was alternatively known as Loculus Archimedius or ‘Archimedes' box’ in Roman times, he was probably not the inventor of the puzzle, but his examinations of its geometrical properties made it a popular toy. Try it yourself, play the game.
Several authors of antiquity have left traces of the puzzle's existence in their work. In the fourth century, the Roman poet Ausonius wrote in his Cento Nuptialis that the Greek called it Ostomachion, a word meaning a fight (μάχη, mákhion) with bones (ὀστέον, ostéon) in reference to the pieces which were often made out of ivory. The Romans themselves knew it as Stomachion, an apparent corruption after a misleading translation into Arabic (where vowels are not written) and confusion with the Greek word for abdominal pain (or stomach ache), which was also used figuratively as meaning “that which drives me mad” (ὀστομάχιον, stomachêin). Whatever the origin of the game and its name, here is a stomach turner.
The first printed reference to the game of tangrams is a book published in China in 1813, although its roots may lie in the furniture used during the Song dynasty in the 12th century. The history of the stomachion puzzle is not only more than twice as long, it is surrounded by mystery every step of the way. When Archimedes died in 212 BC at the hands of a rookie Roman soldier, his writings were considered timeless already, but the Stomachion treatise has been the least understood and was almost lost. For ages it was known only from indirect references and incomplete translations into Arabic and Latin. In the 20th century new texts in the original Greek language were discovered in the Archimedes Palimpsest, and they contain words that lead to the idea that his main concern was not the creation of new forms as previously believed, but the question in how many ways these pieces can be combined back into a square. There is not enough remaining text to be entirely sure, but if this is the case, Archimedes anticipated the field of combinatorics long before that science came into its own during the age of the computer.
Many good websites are devoted to the palimpsest, the puzzle and the person behind them. This site aims to deepen and further broaden the understanding of the Stomachion puzzle and facilitate a way to see all solutions with and without context in a format that is completely customizable to suit many different environments. The Stomachion generator is a PHP-script that accepts multiple parameters for sizes, colours, text overlays and an autorefresh feature. Now anyone can have a daily stomachion on their blog!
The square can be divided in twelve equal parts horizontally and vertically to find the positions of the intersecting lattice points of the dividing lines. Strangely, the corners of the puzzle pieces fall on these points in every configuration. This is certainly not the case for every dissection of a square, but it is unclear whether Archimedes was attracted by this interesting property. In the traditionally known text, he calculates how the sizes of the pieces relate to the whole, and which angles fit together to make a straight line.
There are three sets of pieces which can only go together in the creation of new squares, because they have unique angles which cannot be compensated by other pieces, so they are usually replaced straightaway with single pieces when determining the combinatorical properties of the puzzle. The pair in the top right corner becomes the combined yellow piece in the coloured stomachion on this page, the two very small triangles bottom left become one of the lightblue pieces, and the two longer triangles on the left form the orange triangle. This way the problem is reduced to only eleven pieces.
The remaining text fragments do not reveal whether Archimedes' calculations arrived at the correct answer, or, for that matter, any at all. Only in October 2003, after the palimpsest research had stirred a buzz in the mathematics community, the issue was finally settled by mathematician, puzzle maker and computer programmer Bill Cutler, who wrote a piece of software that automatically scanned all promising constellations for squares. A few weeks later, a team of Californian university professors, including Fan Chung and Ron Graham, provided a theoretical framework to get a grip on the plethora of solutions, supporting his conclusion. It turns out there are 536 unique stomachion squares. If derivations such as rotated and flipped versions are counted separately, the number adds up 32-fold to 17152. If the lightblue and the combined lightblue triangles are considered exchangeable, the number is halved to 268.
The pieces have been ordered according to their relative sizes, an arbitrary choice. The area is given as a fraction of the whole square. The angles in degrees are rounded to full numbers, because some contain an irrational fraction even when expressed in radians proportional to π (pi). In the following table γ (gamma) represents a 45° angle, or ¼π radians. The angle δ (delta) of 26.565051...° is the one whose arctangent is ½.
|A||tetragon||9/48=3/16||63+90+90+117°||2γ - δ, 2γ, 2γ, 2γ + δ||union of two pieces|
|B||triangle||8/48=1/6||27+45+108°||δ, γ, 3γ - δ||union of two pieces, larger version of pieces I and J|
|C||pentagon||7/48||72+135+90+90+153°||γ + δ, 3γ, 2γ, 2γ, 4γ - δ|
|D||tetragon||4/48=1/12||27+126+72+135°||δ, 4γ - 2δ, γ + δ, 3γ|
|E||triangle||"||45+63+72°||γ, 2γ - δ, γ + δ||copy of F|
|F||"||"||"||"||copy of E|
|G||"||3/48=1/16||27+63+90°||δ, 2γ - δ, 2γ||union of two pieces, copy of H|
|H||"||"||"||"||copy of G after the merger|
|I||"||2/48=1/24||27+45+108°||δ, γ, 3γ - δ||copy of J, smaller version of triangle B|
|J||"||"||"||"||copy of I, smaller version of triangle B|
|K||"||"||"||18+54+108°||γ - δ, 2δ, 3γ - δ||the only piece whose sides all have fractional lengths|
By reversing or rotating a symmetrical subset of pieces, one square turns into another. Repeating with another segment then gives a third square. Continuing this way, it is possible to travel through 266 distinct squares. The remaining 2 form a miniloop on their own. This amazing property of the stomachion is only vaguely understood, but the study of graphs and Hamiltonian cycles, as they are called in mathematical jargon, allows some logic to be applied. There is a collection of 24 arrangements, the core squares forming a Hamiltonian path on their own, that are each made up of four triangular subsets of pieces and from which most others can be deducted. Possible shapes for the flipped region are isosceles triangles, trapeziums, parallelograms and rectangles.
Many different paths could have been chosen, this line up has all the logic and surprises one might have come to expect from the stomachion. The ‘default’ arrangement of the pieces, described in the classical texts and commonly used in representations of the puzzle, appears as number 205, on its side and in reverse.
Every unique configuration can be rotated in four directions, and each of those configurations can be mirrored. Thus the number of arrangements is increased by a factor of 4 * 2 = 8.
Four mirrored rotations.
Exchanging two identical triangles doubles the result, and as there are three such pairs of triangular twins (seen here in lightgreen, lightblue and darkblue colours, labeled large, medium and small below), the number is doubled twice more. And 2 * 2 * 2 = 8.
Six exchanged twins.
This makes a grand total of 8 * 8 * 268 = 17152 solutions for the stomachion puzzle.
A system has been devised to extend the numbering to include all 17152 solutions. With it, the generator can be instructed to generate any square. In this scheme, number 269 is the same as number 1 in the Hamiltonian path above, but rotated 90° to the right (clockwise), number 270 is square 2 rotated right 90°, number 3 rotates into 271, number 4 into 272 and so on. The next series, which starts at 268 × 2 + 1 = 537, is the same again, but this time rotated 180°, the one after that is the fourth rotation at 90° to the left (counterclockwise). The next four series are the mirrored versions of the first four, so the flipped (vertically mirrored) version appears in the seventh series as rotated 180° mirrored (orientation 6 if the first is labeled 0). The next eight series are the same, except for the switching of the large twinned triangles. The next sixteen series are the same again, but with exchanged medium triangles. And so on. In the established order of combinations, the large pair is switched every other series (variations 1, 3, 5 and 7 if the first is labeled 0), the medium pair in the second half of every half (2, 3, 6 and 7) and the small pair in the complete second half (4, 5, 6 and 7).
All 64 combinations are listed below. The series number can be used to find the ID of any square. It is multiplied by 268 and added to the ID of the unrotated, unvaried counterpart square in the basic series of the first 268 constellations. The series number is made up of the orientation number and eight times the variation number as shown.
|8×0+0=00||1 - 268||-||-||-||-||-|
|8×0+1=01||269 - 536||right||-||-||-||-|
|8×0+2=02||537 - 804||round||-||-||-||-|
|8×0+3=03||805 - 1072||left||-||-||-||-|
|8×0+4=04||1073 - 1340||-||yes||-||-||-|
|8×0+5=05||1341 - 1608||right||yes||-||-||-|
|8×0+6=06||1609 - 1876||round||yes||-||-||-|
|8×0+7=07||1877 - 2144||left||yes||-||-||-|
|8×1+0=08||2145 - 2412||-||-||yes||-||-|
|8×1+1=09||2413 - 2680||right||-||yes||-||-|
|8×1+2=10||2681 - 2948||round||-||yes||-||-|
|8×1+3=11||2949 - 3216||left||-||yes||-||-|
|8×1+4=12||3217 - 3484||-||yes||yes||-||-|
|8×1+5=13||3485 - 3752||right||yes||yes||-||-|
|8×1+6=14||3753 - 4020||round||yes||yes||-||-|
|8×1+7=15||4021 - 4288||left||yes||yes||-||-|
|8×2+0=16||4289 - 4556||-||-||-||yes||-|
|8×2+1=17||4557 - 4824||right||-||-||yes||-|
|8×2+2=18||4825 - 5092||round||-||-||yes||-|
|8×2+3=19||5093 - 5360||left||-||-||yes||-|
|8×2+4=20||5361 - 5628||-||yes||-||yes||-|
|8×2+5=21||5629 - 5896||right||yes||-||yes||-|
|8×2+6=22||5897 - 6164||round||yes||-||yes||-|
|8×2+7=23||6165 - 6432||left||yes||-||yes||-|
|8×3+0=24||6433 - 6700||-||-||yes||yes||-|
|8×3+1=25||6701 - 6968||right||-||yes||yes||-|
|8×3+2=26||6969 - 7236||round||-||yes||yes||-|
|8×3+3=27||7237 - 7504||left||-||yes||yes||-|
|8×3+4=28||7505 - 7772||-||yes||yes||yes||-|
|8×3+5=29||7773 - 8040||right||yes||yes||yes||-|
|8×3+6=30||8041 - 8308||round||yes||yes||yes||-|
|8×3+7=31||8309 - 8576||left||yes||yes||yes||-|
|8×4+0=32||8577 - 8844||-||-||-||-||yes|
|8×4+1=33||8845 - 9112||right||-||-||-||yes|
|8×4+2=34||9113 - 9380||round||-||-||-||yes|
|8×4+3=35||9381 - 9648||left||-||-||-||yes|
|8×4+4=36||9649 - 9916||-||yes||-||-||yes|
|8×4+5=37||9917 - 10184||right||yes||-||-||yes|
|8×4+6=38||10185 - 10452||round||yes||-||-||yes|
|8×4+7=39||10453 - 10720||left||yes||-||-||yes|
|8×5+0=40||10721 - 10988||-||-||yes||-||yes|
|8×5+1=41||10989 - 11256||right||-||yes||-||yes|
|8×5+2=42||11257 - 11524||round||-||yes||-||yes|
|8×5+3=43||11525 - 11792||left||-||yes||-||yes|
|8×5+4=44||11793 - 12060||-||yes||yes||-||yes|
|8×5+5=45||12061 - 12328||right||yes||yes||-||yes|
|8×5+6=46||12329 - 12596||round||yes||yes||-||yes|
|8×5+7=47||12597 - 12864||left||yes||yes||-||yes|
|8×6+0=48||12865 - 13132||-||-||-||yes||yes|
|8×6+1=49||13133 - 13400||right||-||-||yes||yes|
|8×6+2=40||13401 - 13668||round||-||-||yes||yes|
|8×6+3=51||13669 - 13936||left||-||-||yes||yes|
|8×6+4=52||13937 - 14204||-||yes||-||yes||yes|
|8×6+5=53||14205 - 14472||right||yes||-||yes||yes|
|8×6+6=54||14473 - 14740||round||yes||-||yes||yes|
|8×6+7=55||14741 - 15008||left||yes||-||yes||yes|
|8×7+0=56||15009 - 15276||-||-||yes||yes||yes|
|8×7+1=57||15277 - 15544||right||-||yes||yes||yes|
|8×7+2=58||15545 - 15812||round||-||yes||yes||yes|
|8×7+3=59||15813 - 16080||left||-||yes||yes||yes|
|8×7+4=60||16081 - 16348||-||yes||yes||yes||yes|
|8×7+5=61||16349 - 16616||right||yes||yes||yes||yes|
|8×7+6=62||16617 - 16884||round||yes||yes||yes||yes|
|8×7+7=63||16885 - 17152||left||yes||yes||yes||yes|
|0||No rotation, no reflection|
|1||Rotated 90° right, no reflection|
|2||Rotated 180°, no reflection|
|3||Rotated 90° left, no reflection|
|4||No rotation, reversed|
|5||Rotated 90° right, reversed|
|6||Rotated 180°, reversed|
|7||Rotated 90° left, reversed|
|3||Large and medium pairs|
|5||Large and small pairs|
|6||Medium and small pairs|
|7||All three pairs|
Roman poet Decimus Magnus Ausonius (ca. 310 – ca. 393) mentions the Stomachion in his description of a type of poetry in which various meters are put together, like the pieces of the puzzle, in the introduction to his book Cento Nuptialis (A Wedding Cento):
(Translation based on the one by Hugh G. Evelyn White, published London, 1919. The word ‘cento’ is also Latin for ‘patchwork’.)
The Stomachion has survived the millenia by little more than chance, it seems. The continuous cultivation of interest in and understanding of its mathematical contents, needed for its sustainment, has been threatened by warfare, decline and barbarism more than once. Archimedes wrote his treatises in the form of letters, mostly to his friends in Alexandria. When the library of Alexandria was burned and the school closed, somehow its surviving inheritance was shipped to Byzantium, but when and by who this was done is open to debate. There it appears that a book containing treatises by Archimedes was copied in the sixth century during the time of Isidore of Miletus, and once again in the ninth century thanks to Leo the Geometer. This copy ended up in Italy where it was translated into Latin in the thirteenth century. Now lost, it is known as Codex A. Late in the tenth century, judging by the script, another copy was made. This is the Archimedes Palimpsest, technically known as Codex C. Another Byzantine book, Codex B, containing treatises about optics and mechanics, was also available to the translator of the thirteenth century, but is now lost as well. We are left with copies of copies, and Codex C. All we know today about Archimedes was known and appreciated in the Byzantine world around the new millenium, and probably more. Two centuries later however, it had all been swept away.
|ca. 350 BC||Xenocrates of Chalcedon (Ξενοκράτης ο Χαλκηδόνιος) considers a combinatorical problem when he calculates that
the letters of the Greek alphabet allow the formation of 1002000000000 syllables.
|ca. 287 BC||Archimedes born in Syracuse, an independent Silician Greek city-state in the line of fire between Carthage and Rome.|
|Youth||Probably studies at the Mouseion of Alexandria, famous for its great library, learning from pupils of Euclid.|
|Later life||Invents defense systems for Syracuse, Archimedean screw, writes letters on papyrus about the displacement of
water, the ratio of volume of a cilinder compared to a sphere, the calculation of π (pi) and and and Stomachion.
|Autumn 212 BC||Archimedes killed by a Roman soldier, his books and instruments carried to Rome.|
|48 BC||Library of Alexandria accidentally burned down by Julius Ceasar, according to Plutarch and others.|
|1st century||Roman poet Bassus writes De metris, saying the stomachion game helps children train their memory.|
|2nd century||Introduction of the parchment codex replacing papyrus scrolls.|
|4th century||Roman poet Caius Marius writes in Ars Grammatica there are as many meter types in poetry as stomachion figures.|
|11 May 330||Byzantium proclaimed capital of the Eastern Roman Empire by emperor Constantine the Great, whose name stuck.|
|ca. 370||Roman poet Ausonius composes his Cento Nuptialis for emperor Valentinianus, mentioning the stomachion as a game
of creating different shapes.
|392||Serapeum library in Alexandria, successor of the Mouseion, destroyed by a Christian mob after Emperor Theodosius I
orders the destruction of pagan temples, according to Gibbon.
|425||Pandidakterion (Πανδιδακτήριον) founded in Constantinople by Eastern Roman emperor Theodosius II as a center of
learning in 15 Latin and 16 Greek faculties.
|439||Mar Saba monastery founded in Palestine by Saint Sabbas the Sanctified.|
|ca. 510 onwards||Eutocius of Ascalon writes commentaries on Archimedes and Apollonius, which he dedicates to Anthemius of Tralles.|
|532 - 537||Hagia Sophia ('Holy Wisdom', from the Greek Ἁγία Σοφία), built in Constantinople by Anthemius of Tralles and Isidore
of Miletus, who taught Archimedes in Alexandria and Constantinople, as a tribute On the Sphere and the Cylinder,
would remain the largest cathedral in the world until the completion of Seville Cathedral in 1520.
|Early 800's||Stomachion and other works translated into Arabic by astronomer Thābit ibn Qurra (ثابت بن قرة بن مروان) in Baghdad.|
|9th century||Introduction of the miniscule (lowercase) letter form in manuscript production.|
|850's||Leo the Geometer appointed head of the University of Constantinople, who is remembered in a complimentary note
added to the end of Quadrature of the Parabola in codex A.
|ca. 975||Circa fourth or fifth copy made of Archimedes' writings, now on goatskin parchment but maintaining the format of
columns used on papyrus scrolls, most likely in Constantinople, certainly in the Byzantine Empire.
|1187||Gerardus Cremonensis translates On the Sphere and the Cylinder from Arabic into Latin in Toledo.|
|12 April 1204||Constantinople sacked during the Fourth Crusade, its libraries burned and perhaps all but three books on Archimedes
destroyed, now known as codex A and B, which are taken to Italy as spoils of war, and C, which disappears.
|ca. 1229||Some scientific books taken apart, centuries old text scraped off, euchologion written on top, with prayers typical of
the Jerusalem region.
|18 February 1229||Jerusalem obtained by treaty by the Holy Roman Emperor Frederick II of Germany from Muslim Sultan Malik al-Kamil
during the Sixth Crusade.
|18 March 1229||Frederick II crowned King of Jerusalem in the Church of the Holy Sepulchre.|
|14 April 1229||Euchologion dedicated and signed by ‘presbyter Ioannes Myronas’ on Easter Saturday.|
|ca. 1230 - 1830||Used in services at the Mar Saba monastery near Betlehem, gathering candle wax, losing several pages at the end.
The Stomachion introduction is now the last page.
|15 July 1244||Jerusalem razed by Karizimian Tartars, who were invited by the Sultans to recapture it.|
|1269||Parts of codex A and B, not including Stomachion, translated into Latin by Dominican monk Willem van Moerbeke at
the pontifical court of Viterbo, 100 km north of Rome.
|1311||Codex B last seen in the papal library in Viterbo.|
|ca. 1458||Codex A translated into Latin by Jacobus Cremonensis for Pope Nicholas V.|
|29 May 1453||Constantinople, capital of the Byzantine Empire, captured by Sultan Mehmed II of the Ottoman Empire.|
|ca. 1449 - 1468||Codex A copied into codex E in Venice, now in the Biblioteca Marciana.|
|ca. 1468||Cremonensis copied by German mathematician Regiomontanus with help of codex E.|
|1491||Codex A copied into codex D in Venice for the Biblioteca Laurenziana in Florence after a tip from Angelo Poliziano.|
|16th century||Mar Saba ex libris added to codex C.|
|1503||First publication in print of Archimedes' works, Measurement of a Circle and Quadrature of the Parabola, by
astronomer Luca Gauricio in Venice.
|1544||First complete edition printed by Thomas Gechauff Venatorius in Basel, with the Greek text based on a Nürnberg
manuscript copied from A earlier in the century but with interpolations from B, and the Latin from Regiomontanus.
|1544||Codex A copied into codex H by Christopherus Auverus for the library of Francois I in Fontainebleau.|
|1564||Codex A last seen when its owner, Cardinal Rodolfo Pio da Carpi, dies and his collection passes to the Biblioteca
Estense in Modena, 550 km north of Rome.
|1625||Mar Saba monastery bought by the Patriarch of Jerusalem.|
|Circa 1830's||Mar Saba library collection moved to the Library of the Patriarchate of Jerusalem, and then to the Metochion
("daughter") of the Holy Sepulchre, owned by the Patriarchate, in Constantinople.
|Early 1840's||Codex C noted by German biblical scholar Constantin Tischendorf in the Metochion as an "interesting palimpsest
containing some mathematics".
|1899||Cataloged by Greek paleographer Athanasios Papadopoulos-Kerameus in the Metochion, who copies some of the
mathematical undertext, which German scholar H. Schöne will bring to the attention of Archimedes expert Heiberg.
|1899||Swiss orientalist Heinrich Suter publishes a German translation of Stomachion from a 17th century Arabic manuscript.|
|1906||Palimpsest studied and photographed by Danish philologist Johan Heiberg, identified as unique source of
On Floating Bodies, The Method and Stomachion in original Greek, making headlines the following year.
|1908||Palimpsest last seen by Heiberg.|
|1910 - 1915||Transcription by Heiberg published, uncovering circa 80% of the text.|
|Early 1920's||Metochii Constantinopolitani manuscripts moved to National Library in Athens, although several never arrive.|
|ca. 1923||Palimpsest "bought from a monk" by French businessman Marie Louis Sirieix on a trip to the Orient.|
|1932||Parisian art dealer Salomon Guerson, born 1872 in Turkey, has a manuscript identified by the curator of the
Huntington Library in Los Angeles as the "manuscript of Archimedes described by J.L. Heiberg".
|1934||Guerson offers his manuscript for sale for $6,000.|
|After 1937||Forgeries painted on top of four leaves, covering part of The Method but not Stomachion, containing a green
pigment which first became available in Germany in 1938.
|1942||Guerson escapes from Paris and relocates back to Istanbul.|
|16 July 1942||First roundup of Jews, especially foreign-born, by the occupying Nazi regime in Paris.|
|Around 1940's||Palimpsest stored in unhealthy damp conditions, gathering mold.|
|1956||Palimpsest inherited by Anne Guersan after her father Sirieix dies.|
|1960's||Inspected by philologer Jean Bollack of Lille University after Guersan grows concerned about the book's state.|
|1970's||Inspected by classicist Abraham Wasserstein in Leicester and Joseph Paramelle of the Parisian Institut de Recherche
et d'Histoire des Textes, after which it is restored by Etablissement Mallet in Paris with PVA glue and Blu-Tack.
|1971||A manuscript owned by Cambridge University is identified by Oxfordian classicist Nigel Wilson as a leaf missing from the
palimpsest transcribed by Heiberg, and is subsequently traced back to the estate of Tischendorf.
|1980's||Palimpsest offered for sale to numerous libraries and museums around the world.|
|1991||Offered for auction to Christie's, who appraises it at around 1 million dollars.|
|1998||Examined by Nigel Wilson and photographed under UV-light by Roger Easton of RIT for Christie's sale catalogue.|
|October 1998||Greek patriarch and government try to prevent sale, claiming it was stolen, but lose in court.|
|29 October 1998||Palimpsest sold by Sirieix's grandson Robert Guersan at auction by Christie's in New York to American philantropist
"Mr. B." for $2,202,500, bidding against the Greek general consul for his ministry of Culture.
|15 January 1999||Lent to Walters Art Gallery in Baltimore for conservation and research, where three pages seen by Heiberg are found
to be missing.
|1999 - 2003||Disassembled at an average rate of one leaf per fifteen days.|
|2000||Heiberg's photographs, separated from his other papers and believed lost, found in the Royal Library in Copenhagen.|
|2000 - 2006||Imaging experiments at the Rochester Institute of Technology and John Hopkins University.|
|31 October 2003||Stomachion puzzle solved by Bill Cutler and his computer.|
|November 2003||Stomachion graph theory published by Fan Chung and Ron Graham.|
|2005 - 2006||Imaging experiments at Stanford Synchroton Radition Laboratory.|
|29 October 2008||Tenth anniversary of the research, dataset released, uncovering circa 95% of the text.|
The titles for the different writings of Archimedes have been established by the early scribes, allowing for variations among copies and translations. The parabola was thus named by Apollonius in the 1st century AD. Archimedes used the old term 'section of a right-angled cone'.