In trying to prove that irrational numbers are countable, George Cantor found proof that they are not. Aside from the (enormous) distances, when attempting to build last week's hypothetical “Egyptian Tetrahedron” I found a simple and very visual/obvious demonstration of its impossibility. In fact, the easiest way to construct this tetrahedron (mentally) would be to start from a 3×4 rectangle and fold it along a diagonal until the opposite vertices are 5 units apart. But the point is that they are already at that distance (since the diagonal measures exactly 5 units), which decreases as the rectangle is doubled, which (de)shows that the alleged Egyptian tetrahedron is a flat figure and that the Faces of an equihedral tetrahedron can only be acute triangles. The impossible Egyptian tetrahedron is the limit of the progressive flattening of the equisurface tetrahedron, since one of the angles of its faces tends to 90°.
triangular tetrahedronCarlo Frabetti
What exists is the triangular tetrahedron or trirectangular tetrahedron, in which the three angles of the faces converging at a vertex are correct. The three edges that converge at this vertex are therefore the legs of these faces, which are obviously right triangles, and the three hypotenuses are the sides of the largest face of the triangular tetrahedron, called the base (regardless of position). of the tetrahedron). ). Can you find the height perpendicular to the base of the triangular tetrahedron based on its three legs? And the volume? And the floor space?
Carlo Frabetti
And as a highlight, an elegant problem proposed by Salva Fuster: given an isosceles tetrahedron whose faces are isosceles triangles, find the minimum volume of the sphere containing it, knowing that the volume of the tetrahedron is an integer is.
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Problematic decisions
In recent issues we have seen some problems and paradoxes related to decision theory (see Elsberg's paradox and Simpson's paradox), so it is worth remembering Frank R. Stockton's famous story “The Lady or the Tiger?” which was published in 1882 and has since been often cited when discussing decision-making and free will. In a nutshell the story goes like this:
The protagonist has to choose between two doors: behind one there is a hungry tiger and behind the other there is a beautiful young woman whom he must marry. The protagonist's lover knows which door the tiger is behind. He doesn't want his lover to be devoured by the beast, but he also can't bear the idea of seeing him married to his beautiful rival, and he knows it. She uses a sign to indicate which door he should open. What does he do?
Inspired by this disturbing story, Raymond Smullyan, the great contemporary master of logical puzzles, published a hundred years later – in 1982 – a delicious book with the same title, in which one has to pass many tests as dangerous as the following:
There are two dors. On the I there is a sign that says: “Behind one of these two doors there is at least one lady.” In II there is a sign that says: “Behind the other door there is a tiger.” Which door would you choose if you know that either the two characters are telling the truth or both are lying?
There are three doors. The I sign says: “Behind this door there is a tiger.” In II it says: “Behind this door there is a lady.” In III it says: “Behind door II there is a tiger.” Which door would you choose? if you know that at most one of the three characters is telling the truth?
In both cases it is assumed that you prefer the queen to the tiger.
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