
Hippocrates discovered three squarable lunes. Two more were discovered by Martin Johan Wallenius in 1766. A proof that these are the only squarable lunes was given by Tchebatorew and Dorodnow, and summarized by the famous topologist Postnikov:
• M. M. Postnikov, The problem of squarable Lunes, translated from the Russian by Abe Shenitzer, American Mathematical Monthly, 107 (Aug.-Sep. 2000), 645–651.
However, there may be a loophole in this proof: Will Jagy claims that these Russians assumed without proof that the ratio of two angles involved in the construction of the lune is rational. With this assumption, finding a squarable lune amounts to finding a rational number and a constructible number such that
Puzzle: Why should be rational? Do you know the true state of the art here?
(My puzzles include hard questions I don’t know the answer to, and this is one.)
For a nice discussion of the 5 squarable lunes, with pictures, see this page:
• The five squarable lunes, MathPages.
Twice in a blue lune
Puzzle: Show the area of this right triangle equals the total area inside the blue lunes. The outside of each lune is a semicircle. The inside of each lune is part of the circle containing the points A, B, and C.
The circle through all 3 corners of a triangle is called its circumcircle. You can construct the circumcircle using a straightedge and compass, if you want.
Again this is a famous old problem. The two blue lunes here are called the Lunes of Alhazen. This problem was later posed and solve by Leonardo da Vinci!
This puzzle is a lot of fun, so I urge you not to give up—but if you do, you can see da Vinci’s solution here.
Puzzle: show the area of the green shape equals the area of the circle.
The green shape is called an arbelos, which means ‘shoemaker’s knife’ in Greek, since it looks a bit like that. It shows up in Propositions 4-8 of the Book of Lemmas. This book goes back at least to Thābit ibn Qurra, a mathematician, astronomer and physician who live in Baghdad from 826 to 901 AD. Ibn Qurra said the book was written by Archimedes! But nobody is sure.
So, when you do this puzzle, you may be matching wits with Archimedes. But you’ll certainly be sharing some thoughts with Thābit ibn Qurra.
Math has been called the world’s longest conversation. Perhaps this is an exaggeration: people have been passing on stories for a long time. But Babylonians were computing the square root of two back in 1700 BC, probably using techniques that are still interesting today… so it really is remarkable how old mathematics still makes good sense, unlike the old creation myths.
For more on the arbelos try:
• Arbelos, Wikipedia.
• Thomas Schoch, Arbelos references, 2013.
For the 15 propositions in the Book of Lemmas, see:
• Book of Lemmas, Wikipedia.
Two semicircles is half a circle
Puzzle: show the total area of the two semicircles is half the area of the large circle.
Amazingly, it seems this fact was noticed only in 2011:
• Andrew K. Jobbings, Two semicircles fill half a circle, The Mathematical Gazette 95 (Nov. 2011), 538–540.
However, Andrew Jobbings is a genius when it comes to ‘recreational mathematics’. For more of his work, check out this page:
• Andrew Jobbings, Arbelos.
• Alexander Bogomolny, A property of semicircles.
You’ll get a scary warning asking “Do you want to run this application?” Say yes. You’ll get an applet that lets you slide the point where the semicircles touch: no matter where it is, the semicircles have the same total area! Click “hint” and you’ll get a hint. If you’re still stuck, and too impatient to solve the puzzle yourself, scroll down and see a proof!
However, that proof is long and confusing. With geometry I like demonstrations where after some thought you can simply see that the result is true. For this puzzle, such a demonstration was provided by Greg Egan.
Click here if you give up on this puzzle and want to see Egan’s solution. You may need to ponder it a bit, but when you’re done you should say “yes, this result is clear!“
As a separate hint: the answers to this puzzle and the previous one are similar, in