An Experimental (yet fully rigorous!) Study of a certain "Measure Of Disarray" that 12-year Noga Alon Proved was always Even

By Shalosh B. Ekhad and Doron Zeilberger

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First Written: Sept. 29, 2021. This version: Oct. 14, 2021.

Last update of this web-page (but not article): Dec. 16, 2021

In a beautiful new "coffee table book", "Do not Erase", by the very talented artistic photographer Jessica Wynne, there are pictures of more than one hundred blackboards by a very diverse set of mathematicians. One of them is Noga Alon's blackboard. Each blackboard photo is accompanied by a short essay by the creator of the blackboard, where they often describe how they decided to become mathematicians. According to Noga Alon, the "epiphany" occured when he was 12 years old, when he settled a heated argument in a "Eurovision watching party" that his parents threw, where he conclusively proved that a certain "measure of disarray" must always be even, in particular causing one of the guests, "a grown-up engineer", to concede that he was wrong in claiming that it was a coincdence that the scores turned out to be all even. According to Noga, the fact that a 12-year-old can win an argument against a grown-up (especially one who is an engineer) shows the objectivity of mathematical truth, and reinforced his decision to become a mathematician.

While I do agree that mathematical knowledge is more objective than most other kinds, it is not as objective as it seems. But the point of the present article is to investigate, in a purely empirical (yet fully rigorous!) way, that same measure of disarry, that turned out to be called "Spearman's footrule", going far beyond just proving that it is always even, considerably extending a 1977 paper by Persi Diaconis and Ron Graham, debunking yet-another myth: that mathematics is always deductive.

Here is the talk.

Added Sept. 30, 2021: Stoyan Dimitrov told us about these two interesting papers: