Henry Perigal, London, 1873
Henry Perigal (1801–1898) kept the books in a London stockbroker’s office and spent his spare time cutting shapes apart and fitting them back together. He printed this puzzle in 1873 in a math magazine, the Messenger of Mathematics. Elisha Loomis’s big collection of proofs (1927) lists it as proof 3.
The idea (opens when you solve it)
Two cuts through the middle of the bigger leg square, one of them parallel to the triangle’s longest side, make four matching pieces. They and the small square slide, without turning, into the square on the longest side.
Loomis, The Pythagorean Proposition (1927), geometric proof 3, p. 89. Free scan: HathiTrust
Bhaskara, India, 1100s
Bhaskara (born 1114) was a mathematician and astronomer in India. The story goes that he drew a figure like this one and wrote just one word under it: “Behold!” Loomis (1927) shows his proof in two pictures; this puzzle starts from the second, with the two leg squares side by side.
The idea (opens when you solve it)
The square on the longest side is four copies of the triangle around a little square. The same five pieces, moved around, fill the two leg squares sitting side by side.
Loomis, The Pythagorean Proposition (1927), geometric proof 154, figs. 212a–b, p. 182. Free scan: HathiTrust
Thabit ibn Qurra, Baghdad, 800s
Thabit ibn Qurra (836–901) grew up in Harran and moved to Baghdad, where he worked as a translator, doctor and mathematician. In a short work about an old Greek proof of this theorem, he gave new proofs of his own. This puzzle follows the first of them, run backwards.
The idea (opens when you solve it)
Take two copies of the triangle off the square on the longest side and stick them back on the other sides: the shape you get is exactly the two leg squares together.
Thabit ibn Qurra’s letter, in English in A. Sayili, Isis 51 (1960), 35–37. His life: MacTutor
an-Nayrizi, Baghdad, about 900
Al-Nayrizi (died about 922) came from the town of Nayriz in Persia and worked in Baghdad. He wrote a long commentary on Euclid’s Elements that survives in Arabic and in a Latin translation from the 1100s. His proof starts from a floor pattern of big and small square tiles.
The idea (opens when you solve it)
Lay a grid of tilted squares, each the size of the square on the longest side, over the tile floor. Each tilted square holds exactly the pieces of one small tile and one big tile.
Al-Nayrizi, commentary on Euclid’s Elements, Book I, Proposition 47. More: Pythagorean tiling (Wikipedia)
Four triangles in a square
Nobody knows who first saw this one. An old Chinese book, the Zhoubi suanjing, has a picture added by Zhao Shuang in the 200s AD of four triangles inside a 7-by-7 square with a tilted square in the middle. Today it is one of the most popular proofs of all.
The idea (opens when you solve it)
Four copies of the triangle in a big square leave empty space. Set them one way and the space is the square on the longest side; set them another way and it is the two leg squares. Same big square, same triangles, so the spaces are equal.
The hypotenuse diagram in Zhao Shuang’s commentary on the Zhoubi suanjing. More: Zhoubi suanjing (Wikipedia)
Liu Hui, China, AD 263
In AD 263 Liu Hui wrote notes on an older Chinese book, the Nine Chapters on the Mathematical Art. He wrote that the small square and the big square can be “moved about so as to patch each other” into the square on the longest side, but his picture is lost. This puzzle is one modern guess at it, by the historian Donald Wagner (1985); Benjir von Gutheil (died 1914) had drawn the same cut earlier.
The idea (opens when you solve it)
Cut both leg squares into pieces and fit every piece straight into the square on the longest side, with nothing left over. Two pieces have to be turned over.
D. B. Wagner, “A proof of the Pythagorean theorem by Liu Hui”, Historia Mathematica 12 (1985), 71–73. Free version: donwagner.dk
Two hexagons, Germany, 1700s
This proof is often credited to Leonardo da Vinci, but there is no evidence that he knew it. It was printed in a German geometry book in 1790, and an 1826 book says Johann Tobias Mayer of Göttingen had found it in 1772. Leonardo’s name was first attached to it in 1906. Loomis (1927) lists it as proof 23.
The idea (opens when you solve it)
Each six-sided shape is two copies of the triangle plus squares: the two leg squares in one, the square on the longest side in the other. The halves of the first shape make the second, so what is left over must be equal.
F. Lemmermeyer, “Leonardo da Vinci’s proof of the Theorem of Pythagoras” (2013); Loomis (1927), proof 23, p. 104. Free: arXiv
