To enter 2024 without unfinished business, let's start by seeing the solutions to the riddles of master Raymond Smullyan proposed in the two previous installments.
In the first of the riddles, if sign II is false there is a lady behind door I, and therefore behind at least one door there is a lady, then sign I is true. Therefore, both signs cannot be false, therefore both are true, therefore there is a tiger behind door I and a lady behind door II.
In the second puzzle, signs II and III make mutually exclusive statements, so one of the two is true. And since at most one of the three signs is true, the first one is false, then the lady is behind door I.
As for the third of the riddles, inspired by the story of Edgar Allan Poe Dr. Brea's method and Professor Pluma (also translated into Spanish as The method of Dr. Alquitrán and Professor Trapaza), I will limit myself to giving the solution (the development is too long to include it in its entirety), which those who know Poe's disturbing story will have guessed without having to solve it: in that upside-down madhouse, all the supposed doctors are crazy and all the supposed The patients are sane, because – according to Poe – at one point the crazy people revolted, tarred and feathered the center's staff (hence the names Brea and Pluma) and locked them in the corresponding cells.
The cannonball problem
The number 2024 does not give much use from a mathematical point of view (or does it?), unlike its ending, 24, which has no waste and which continually appears in colloquial language for the mere fact that the day is divided in 24 hours (do you know or can you deduce why?). And although most watches divide their dial into 12 parts for greater convenience, since it is not usually difficult to distinguish between day and night, there are still historical clocks (especially astronomical ones) with the dial divided into 24 sectors.
From a mathematical point of view, 24 is a highly abundant and highly composite number, it is a practical number, it is a Harshad number, it is a refactorable number… And I am sure that my astute readers will find more notable properties in it.
And if that were not enough, 24 is the only non-trivial solution to the famous cannonball problem, which we have dealt with on occasion. The problem raises the question of which numbers are both squares and pyramidal squares. Let us remember that a square pyramidal number corresponds to a pyramidal stacking of spheres with a square base, as ancient cannonballs used to be stacked, so its sequence is 1, 5, 14, 30, 55, 91…
It would be too much to ask you to prove that 24 is the only non-trivial solution (the trivial one is 1, obviously) to the cannonball problem, which was not demonstrated until 1918 by GN Watson—long after being posed by the great French mathematician Édouard. Lucas—using elliptic functions; but it is not too much to ask that you demonstrate that 24 is indeed a solution.
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