ten 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 a very nice way!