# The other bathtub of Archimedes | Science

Certain ambiguity (mea culpa) in the statement of some problem related to the "Drunken walk" (random tours) has led to a broad and interesting debate (See comments of the last two weeks), so, for once, it has been fulfilled that of "there is no harm that does not come well".

On the drunken king's gait in the center of the chessboard, I reproduce the conclusions, after arduous discussions, of our habitual commentator Oli Limón:

Unless error or new contributions, the odds of the king are as follows:

Probability that reaches the edge in 3 runs: 2/43= 1/32 = 0.03125

Probability that reaches the edge in 4 runs or less: 1/32 +18/44= 13/128 = 0.1015625

Probability that reaches the edge in 5 runs or less: 13/128 + 108/45= 57/512 = 0.20703125

Probability return starting point in 4 runs or less: 1/4 + 36/44= 25/64 = 0.390625

As for the endless walk of the drunken king for an unlimited board, I reproduce the comment of another "prominent user", Manuel Amorós:

According to Martin Gardner, the probability in an infinite walk along the reticle, to visit any point of it is 1. In other words, sooner or later, if the walk continues indefinitely, we will return to the origin. Things change radically in a three-dimensional grid, I quote Gardner: "In 1940, McCrea and Whipple showed that the probability of the walk returning to the origin of his walk is only 0.35 (approximately), although the walk lasts indefinitely ".

If I remember correctly, Ian Stewart came to the same conclusion. The demonstration exceeds the limits of this section, but that is the data for those who wish to deepen this interesting and elusive question.

### The distrustful king

And from a drunken king to a distrustful one.

The history of the bathtub of Archimedes (of which we have dealt at some time on this page) is well known, which gave him the idea to calculate the volume of the crown of the distrustful King of Syracuse, Hiero II.

Less known and reliable is the legend of another bathtub, very large and luxurious, which allowed Archimedes to show off for the second time before the distrustful king. Hiero had ordered that they make him a bronze bathtub capable of holding a thousand liters of water (said in current units, obviously), but he had the feeling that it was smaller than agreed, so he asked Archimedes to calculate his capacity.

– For that you send me to call? the wise man complained. Say fill it using a ten-liter vessel and count if you have to use it a hundred times.

"The bathtub is already full of hot water and I do not want to waste it," Hieron replied.

– Then bathe and then empty it using the ten-liter vessel.

-I do not want to use a deceptive bathtub, which would not be worthy of my real person.

How did Archimedes manage to calculate the capacity of the bathtub without emptying it?

And since this installment appears on December 28, I have included in it, as a meta-construct, a little joke that my astute readers will have to discover. (A small clue: the little joke is a small tribute to the great Raymond Smullyan).

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.