## Archive for Norway

## wildlife photograph of the year [The Guardian]

Posted in pictures, Travel with tags British Wildlife Photography Awards, fish, fishing, Norway, oceans, The Guardian, wildlife photography on September 11, 2021 by xi'an## la remise en cause des mathématiques comme outil exclusif de reproduction de la bourgeoisie

Posted in Books, Kids, pictures, University life with tags Baccalauréat, calculus, curriculum, French government, French politics, high school mathematics, history, Le Monde, Norway, reformation on June 18, 2021 by xi'an

**A** tribune that was published by Le Monde a few days ago celebrates the end of the “dominance of mathematics [in high school programs] as the unique reproduction medium of the bourgeoisie”, in connection with a recent reformation of French high school programs where students have to specialise in only three topics in their final years. This change has led to a major drop both in the number of students studying maths and in the contents of the maths curriculum. As a result, there will less students entering university with a basic maths background and an overall regression in their level. At a time when international scores show French pupils are on average the worst ones in Europe and when the French government has huge ambitions to develop national AI companies, this drift should be most concerning… But not for the author of the tribune, a high school professor of history and geography, who is most happy in the rise of students specialising in his subject, with a caricaturesque opinion on the inegalitarian role of mathematics:

“[la réforme] devait dès lors permettre, par le jeu des nouvelles spécialités, l’expression d’aptitudes plus diverses et d’en finir avec la prééminence systématique des mathématiques comme instrument de sélection scolaire et sociale.”[the reformation should then allow through new specialties to account for a wider range of abilities and to end the systemic preeminence of mathematics as a tool for school and social selection]

“[les mathématiques] demeurent le choix privilégié des mâles CSP + soucieux de préserver leur rang social”[mathematics still are the favoured option of higher class males afraid to loose their social position]

“[la spécialité histoire-géographie-sciences politiques] doit contribuer à la promotion sociale des plus défavorisés et à la remise en cause des mathématiques comme outil exclusif de reproduction de la bourgeoisie.”[the history, geography and political science specialty must contribute to the social promotion of the least favoured and to the demotion of mathematics as the unique instrument of preservation of the bourgeoisie]

If it was not so sadly representative of a general perception of mathematics within the global population and among the high administration of the Education Ministry, the outdated ideological tone of the tribune would have been quite hilarious.

## Bernoulli factory in the Riddler

Posted in Books, Kids, R, Statistics with tags Agent Orange, Bernoulli factory, binomial distribution, combinatorics, FiveThirtyEight, John von Neumann, kroner, Norway, polyroot, R, The Riddler on December 1, 2020 by xi'an

“Mathematician John von Neumann is credited with figuring out how to take a p biased coin and “simulate” a fair coin. Simply flip the coin twice. If it comes up heads both times or tails both times, then flip it twice again. Eventually, you’ll get two different flips — either a heads and then a tails, or a tails and then a heads, with each of these two cases equally likely. Once you get two different flips, you can call the second of those flips the outcome of your “simulation.” For any value of p between zero and one, this procedure will always return heads half the time and tails half the time. This is pretty remarkable! But there’s a downside to von Neumann’s approach — you don’t know how long the simulation will last.”The Riddler

**T**he associated riddle (first one of the post-T era!) is to figure out what are the values of p for which an algorithm can be derived for simulating a fair coin in at most three flips. In one single flip, p=½ sounds like the unique solution. For two flips, p²,(1-p)^2,2p(1-p)=½ work, but so do p+(1-p)p,(1-p)+p(1-p)=½, and the number of cases grows for three flips at most. However, since we can have 2³=8 different sequences, there are 2⁸ ways to aggregate these events and thus at most 2⁸ resulting probabilities (including 0 and 1). Running a quick R code and checking for proximity to ½ of any of these sums leads to

[1] 0.2062997 0.7937005 #p^3 [1] 0.2113249 0.7886753 #p^3+(1-p)^3 [1] 0.2281555 0.7718448 #p^3+p(1-p)^2 [1] 0.2372862 0.7627143 #p^3+(1-p)^3+p(1-p)^2 [1] 0.2653019 0.7346988 #p^3+2p(1-p)^2 [1] 0.2928933 0.7071078 #p^2 [1] 0.3154489 0.6845518 #p^3+2p^2(1-p) [1] 0.352201 0.6477993 #p^3+p(1-p)^2+p^2(1-p) [1] 0.4030316 0.5969686 #p^3+p(1-p)^2+3(1-p)p^2 [1] 0.5

which correspond to

1-p³=½, p³+(1-p)³=½,(1-p)³+(1-p)p²=½,p³+(1-p)³+p²(1-p),(1-p)³+2(1-p)p²=½,1-p²=½, p³+(1-p)³+p²(1-p)=½,(1-p)³+p(1-p)²+p²(1-p)=½,(1-p)³+p²(1-p)+3p(1-p)²=½,p³+p(1-p)²+3(p²(1-p)=½,p³+2p(1-p)²+3(1-p)p²=½,p=½,

(plus the symmetric ones), leading to 19 different values of p producing a “fair coin”. Missing any other combination?!

Another way to look at the problem is to find all roots of the equations

where

(None of these solutions is rational, by the way, except p=½.) I also tried this route with a slightly longer R code, calling polyroot, and finding the same 19 roots for three flips, ~~[at least]~~ 271 for four, and ~~[at least]~~ 8641 for five (The Riddler says 8635!). With an imprecision in the exact number of roots due to rather poor numerical rounding by polyroot. (Since the coefficients of the above are not directly providing those of the polynomial, I went through an alternate representation as a polynomial in (1-p)/p, with a straightforward derivation of the coefficients.)