Le Monde [reverse] rank test

This is the fourth and hopefuly last post about this puzzle. If I translate the problem proposed by Le Monde, it reads as follows

Twenty pupils in the class have different grades that are the integers from 1 to 20. The ten girls in the class are ordered from the best grade to the worst one, while the ten boys in the class are placed from the worst grade to the best one. The absolute differences between the pairs thus formed are computed and sum up. What is the range for this sum?

which is different from what I “read”, where both boys and girls were ranked in increasing order. Of course, “my” reading makes more sense (!) from a statistical point of view, because this defines a rank test for both samples having the same distribution. (The range is then between 10 and 100.) However, the solution to the original problem published in the weekend special edition is that the sum is always equal to 100. The argument is that any number less than 10 is paired with a number larger than 10, thus that the numbers larger than 10 get a positive sign, while the numbers less than 10 always get a negative factor, leading to

Obviously, this result holds for any balanced group of pupils. This is however much less interesting from a statistical perspective.

Ps- I found recently that both writers of the “Affaire de Logique” page in the weekend Le Monde magazine, Elisabeth Busser and Gilles Cohen, are in fact editors of a math fanzine called Tangente. Gilles Cohen wrote a laudatory review of the book, Le Mythe Climatique, by Benoît Rittaud, next to an explanation by Benoît Rittaud of the findings of Ed Wegman and of his Academy of Sciences committee about the hockey stick temperature curve. While the problem with the hockey stick is clear enough, the data being recentred only against recent observations, the explanations given in Tangente are fairly obscure. As a coincidence, Benoît Rittaud just decided to put his blog on hold and to move to a collective climatoskeptic blog called skyfall