In previous weeks we have seen the surprising relationship of the Tower of Hanoi with Hamiltonian tours, as well as with the legend of the inventor of chess, and the versatile tower still has some surprises in store. For example, its relationship with the Sierpinski triangle, pointed out by our regular commentator Luca Tanganelli.
Although the Polish mathematician Waclaw Sierpinski (1882-1969) is best known for his famous “carpet”, he also devised other fractal objects, such as the triangle that bears his name, which is obtained as follows:
In an equilateral triangle (although any triangle will do), we join the midpoints of the sides and eliminate the central triangle thus obtained (in white in the figure), leaving 3 triangles whose joint area is 3/4 of that of the initial triangle. We do the same with the three remaining triangles, leaving 9 triangles whose joint area is 9/16 of that of the initial triangle… and so on indefinitely.
Well, as Tanganelli points out: “The graph of the movements of the Tower of Hanoi is like a Sierpinski triangle, and the shortest path between two extreme positions (where the disks are all on one axis) is visualized as one side of said triangle. I asked myself if there was a longer path between two extreme positions, and it turns out there is. This path turns out to be a Hamiltonian path.”
(I clarify that the graph of the trivial tower with a single disk is the initial triangle, that of the tower with two disks is the first step of the Sierpinski process, that is, the triangle divided into four parts, and so on).
As we saw, the shortest path, for a tower of n disks, requires 2ⁿ – 1 moves. How many moves does the Hamiltonian path require? Are both paths unique?
The true name of God
And speaking of curious parallels, the (apocryphal) legend of the monks of Benares who incessantly move the 64 gold discs from a tower in Hanoi, whose enormous task, when completed, will mean the end of the world, has its counterpart in a famous short story by Arthur Clarke titled The nine billion names of Godwhich tells the story of some Tibetan monks who constantly combine the letters of their alphabet to try to form the true name of God; When they find it, nothing will be left to do and the stars will go out. Taking into account that the Tibetan alphabet consists of thirty letters, that the name of God cannot have more than nine letters and that the same letter cannot appear more than three times in a row (as this would result in a name that would be unpronounceable even for a monk Tibetan), is the number of possible divine names really on the order of billions? Or are those who mistakenly translate billions as trillions closer to the truth in this case?
Let us set ourselves a somewhat simpler task than that of the Tibetan monks: suppose that whoever called the supreme being “God” was correct with the number of letters and the proportion of vowels and consonants, but did not find the true name. How many are the possible names with four letters, two vowels and two consonants, compatible with the morphology of Spanish? But don’t recite them out loud, just in case…
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