May 16, 2021

# From Archimedes to Smullyan, passing through Leibniz and Cantor | Science

Archimedes threw 10 kilos of salt in King Hieron's macrobaby (see riddle of last week), which improved the bath water, converting it into a Hippocratic solution (equivalent to physiological saline solution). Once the salt was well dissolved, he took a liter of water from the bathtub and boiled it until all the liquid evaporated and the salt was left, which weighed more than 10 grams, which meant that the capacity of the bathtub was lower to 1,000 liters. The king was once again astonished by the sagacity of Archimedes, who also benefited from a great bath in warm physiological saline (more pleasant and salutary for the skin, mucous membranes and eyes than pure water), as Hieron refused to enter a deceptive bathtub. Presumably the manufacturer of the bathtub suffered the same fate as that of the crown which turned out not to be of pure gold; But that is another story.

As for the "metainocentada" of day 28, and as our "featured user" Manuel Amorós discovered, is directly inspired by the prologue of the delicious book of logical riddles How is this book called?, by maestro Raymond Smullyan. In that prologue, Smullyan tells that, when he was a boy, his brother said to him: "Ray, today I am going to spend a joke that you will never forget". He spent the day waiting for a joke that did not come, and in the end his brother told him that the joke was to make him believe that he was going to play a joke.

### The infinite and more here

The endless gait of a drunken king by an unlimited board provoked an extensive and intense debate that at the time of writing these lines has not yet been extinguished (see comments of the previous two weeks). Chance and infinity are two of the most slippery concepts, in and out of mathematics, and when they go together and mixed up the result can be very disconcerting. The echoes of the confrontation between Cantor and Kronecker, of which we have spoken in these same pages on more than one occasion, still resonate, and at the time the infinitesimal calculation of Leibniz and Newton raised not a few philosophical blisters (as a balm for that type of blisters, I recommend another fascinating Smullyan book: Satan, Cantor and infinity).

Coincidentally (or maybe not), Archimedes was the first to tame the infinite to put it at the service of geometry, anticipating in two thousand years the "indivisible" Cavalieri, of which we did not occupy a year ago (see "The principle of Cavalieri", 1-12-2017), and the infinitesimal calculation itself.

I invite my wise readers to do a little Arquimedian feat: deduce the area of ​​the circle considering it a regular polygon of infinite sides.

And since we just entered the year 2019 and this is the delivery number 187 of "The game of science", what can you say about these two beautiful numbers? Do they have something in common or can they relate to each other in some interesting way?

Carlo Frabetti He is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 scientific dissemination works for adults, children and young people, among them Damn physics,Damn mathematics or The big game. He was a screenwriter The Cristal ball.