In attempting to prove that irrational numbers are countable, George Cantor found proof that they are not. Saving the (enormous) distances, when trying to construct the hypothetical "Egyptian tetrahedron" last week, I found a simple and very visual/obvious demonstration of its impossibility. In fact, the simplest way to (mentally) construct such a tetrahedron would be to start from a 3x4 rectangle and bend it diagonally until the opposite vertices were at a distance of 5 units. But the point is that they are already at that distance (since the diagonal measures precisely 5 units), which will decrease when the rectangle is folded, which (de)shows that the supposed Egyptian tetrahedron is a flat figure, and that the faces of an equihedral tetrahedron can only be acutangle triangles. The impossible Egyptian tetrahedron is the limit of the progressive flattening of the equihedral tetrahedron as one of the angles of its faces tends to 90º.
trirectangle tetrahedronCarlo Frabetti
The one that does exist is the trirectangular tetrahedron or trirectangle tetrahedron, which is the one in which the three angles of the faces that converge at a vertex are straight. The three edges that converge at that vertex are thus the legs of these faces, which are obviously right triangles, and the three hypotenuses are the sides of the major face of the trirectangle tetrahedron, which is called the base (regardless of the position of the tetrahedron). Can you find the height perpendicular to the base of the trirectangle tetrahedron based on its three legs? What about volume? What about the base area?
Carlo Frabetti
And to top it off, an elegant problem proposed by Salva Fuster: given an equihedron tetrahedron whose faces are isosceles triangles, find the minimum volume of the sphere that contains it, knowing that the volume of the tetrahedron is an integer.
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Napoleon's Theorem
Problematic Decisions
In recent installments 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?, published in 1882 and since then often cited when talking about decision-making and free will. Very briefly, the story is as follows:
The protagonist has to choose between two doors: behind one of them there is a hungry tiger and behind the other a beautiful young woman whom he will have to marry. The protagonist's lover knows which door the tiger is behind. He doesn't want to see his beloved devoured by the beast, but he can't bear the thought of seeing him married to his beautiful rival, and he knows it. She signals him which door to open. What does he do?
Inspired by this disturbing tale, Raymond Smullyan, the great contemporary master of logical puzzles, published a hundred years later – in 1982 – a delightful book with the same title, in which you have to overcome a lot of tests as dangerous as the following:
There are two doors. On the I is a sign that reads: "At least behind one of these two doors is a lady." On the second there is a sign that says: "There is a tiger behind the other door". Knowing that either the two signs are telling the truth or they are both lying, which door would you choose?
There are three doors. The sign on the I reads, "Behind this door is a tiger." The one in the second reads: "Behind this door there is a lady." The one in Gate III reads: "Behind Door II there is a tiger." Knowing that at most one of the three signs tells the truth, which door would you choose?
In both cases you are supposed to prefer the lady to the tiger.
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