Le Monde puzzle [#1127]

A permutation challenge as Le weekly Monde current mathematical puzzle:

When considering all games between 20 teams, of which 3 games have not yet been played, wins bring 3 points, losses 0 points, and draws 1 point (each). If the sum of all points over all teams and all games is 516, was is the largest possible number of teams with no draw in every game they played?

The run of a brute force R simulation of 187 purely random games did not produce enough acceptable tables in a reasonable time. So I instead considered that a sum of 516 over 187 games means solving 3a+2b=516 and a+b=187, leading to 142 3’s to allocate and 45 1’s. Meaning for instance this realisation of an acceptable table of game results

games=matrix(1,20,20);diag(games)=0
while(sum(games*t(games))!=374){
  games=matrix(1,20,20);diag(games)=0
  games[sample((1:20^2)[games==1],3)]=0}
games=games*t(games)
games[lower.tri(games)&games]=games[lower.tri(games)&games]*
    sample(c(rep(1,45),rep(3,142)))* #1's and 3'
    (1-2*(runif(20*19/2-3)<.5)) #sign
games[upper.tri(games)]=-games[lower.tri(games)]
games[games==-3]=0;games=abs(games)

Running 10⁶ random realisations of such matrices with no constraint whatsoever provided a solution with] 915,524 tables with no no-draws, 81,851 tables with 19 teams with some draws, 2592 tables with 18 with some draws and 3 tables with 17 with some draws. However, given that 10*9=90 it seems to me that the maximum number should be 10 by allocating all 90 draw points to the same 10 teams, and 143 3’s at random in the remaining games, and I reran a simulated annealing version (what else?!), reaching a maximum 6 teams with no draws. Nowhere near 10, though!

Le Monde puzzle [#1120]

A board game as Le weekly Monde current mathematical puzzle:

11 players in a circle and 365 tokens first owned by a single player. Players with at least two tokens can either remove one token and give another one left or move two right and one left. How quickly does the game stall, how many tokens are left, and where are they?

The run of a R simulation like

od=function(i)(i-1)%%11+1
muv<-function(bob){
  if (max(bob)>1){
    i=sample(rep((1:11)[bob>1],2),1)
    dud=c(0,-2,1)
    if((runif(1)<.5)&(bob[i]>2))dud=c(2,-3,1)
    bob[c(od(i+10),i,od(i+1))]=bob[c(od(i+10),i,od(i+1))]+dud
  }
  bob}

always provides a solution

> bob
 [1] 1 0 1 1 0 1 1 0 1 0 0

with six ones at these locations. However the time it takes to reach this frozen configuration varies, depending on the sequence of random choices.

 

Deirdre McCloskey dans Le Monde

Michael dans le Monde [#2]

A (second) back-page interview of Mike in Le Monde on the limitations of academics towards working with major high tech companies. And fatal attractions that are difficult to resist, given the monetary rewards. As his previous interview, this is quite an interesting read (in French), although it obviously reflects a US perspective rather than French (with the same comment applying to the recent interview of Yann LeCun on France Inter).

“…les chercheurs académiques français, qui sont vraiment très peu payés.”

The first part is a prediction that the GAFAs will not continue hiring (full-time or part-time) academic researchers to keep doing their academic research as the quest for more immediate profits will eventually win over the image produced by these collaborations. But maybe DeepMind is not the best example, as e.g. Amazon seems to be making immediate gains from such collaborations.

“…le modèle économique [de Amazon, Ali Baba, Uber, &tc] cherche à créer des marchés nouveaux avec à la source, on peut l’espérer, de nouveaux emplois.”

One stronger point of disagreement is about the above quote, namely that Uber or Amazon indeed create jobs. As I am uncertain that all jobs creations are worthwhile. Indeed, which kind of freedom there is in working after-hours for a reward that is so much below the minimal wage (in countries where there is a true minimal wage) that the workers [renamed entrepreneurs] are below the poverty line? Similarly, unless there are stronger regulations imposed by states or unions like the EU, it seems difficult to imagine how society as an aggregate of individuals can curb the hegemonic tendencies of the high tech leviathans…?

the 101 favourite novels of Le Monde readers

Le Monde called its readers to vote for their five favourite novels, with no major surprise in the results, except maybe Harry Potter coming up top. Before Voyage au bout de la nuit and (the predictable) A la recherche du temps perdu. And a complete unknown, Damasio’s La Horde du Contrevent, as 12th and first science fiction book. Above both the Foundation novels (16th). And Dune (32nd). And Hyperion Cantos (52). But no Jules Verne! In a sense, it reflects upon the French high school curriculum on literature that almost uniquely focus on French 19th and 20th books. (Missing also Abe, Conrad, Chandler, Dickens, Ishiguro, Joyce, Kawabata, Madame de Lafayette, Levi, Morante, Naipaul, Rabelais, Rushdie, Singer, and so many others…) Interestingly (or not), Sartre did not make it to the list, despite his literature 1953 Nobel Prize, maybe because so few read the (appalling) books of his chemins de la liberté trilogy.

I did send my vote in due time but cannot remember for certain all the five titles I chose except for Céline’s Voyage au bout de la nuit (2nd), Cormac McCarthy’s The Road (74th) and maybe Fedor Dostoievski’s Brothers Karamazov (24th). Maybe not as I may have included Barbey d’Aurevilly’s L’ensorcelée, Iain Pears’ An instance at the fingerpost, and Graham Greene’s The End of the affair, neither of which made it in the list. Here are some books from the list that would have made it to my own 101 list, although not necessarily as my first choice of titles for authors like Hugo (1793!) or Malraux (l’Espoir). (Warning: Amazon Associate links).

• Harry Potter
• Voyage au bout de la nuit
• One hundred years of solitude
• The Lord of the rings
• L’étranger
• Belle du seigneur
• Les misérables
• Mémoires d’Hadrien
• Pride and prejudices
• The unbearable lightness of being
• Foundation
• Crime and punishment
• Madame Bovary
• Le rouge et le noir
• The Brothers Karamazov
• La peste
• War and peace
• The grapes of wrath
• Les liaisons dangereuses
• Bel-Ami
• Anna Karenina
• The name of the rose
• Germinal
• To kill a mocking bird
• The three musketeers
• Kafka on the shore
• La vie mode d’emploi
• Lolita
• Wuthering heights
• Mort à crédit
• The picture of Dorian Gray
• Life and fate
• Chess story
• La chartreuse de Parme
• L’œuvre au noir
• The idiot
• L’assommoir
• The road
• Le père Goriot
• The trial
• Don Quixote
• Notre-Dame de Paris
• Illusions perdues
• The sound and the fury
• La condition humaine

Le Monde puzzle [#1124]

A prime number challenge [or rather two!] as Le weekly Monde current mathematical puzzle:

When considering the first two integers, 1 and 2, their sum is 3, a prime number. For the first four integers, 1,2,3,4, it is again possible to sum them pairwise to obtain two prime numbers, eg 3 and 7. Up to which limit is this operation feasible? And how many primes below 30,000 can write as n^p+p^n?

The run of a brute force R simulation like

max(apply(apply(b<-replicate(1e6,(1:n)+sample(n)),2,is_prime)[,b[1,]>2],2,prod))

provides a solution for the first question until n=14 when it stops. A direct explanation is that the number of prime numbers grows too slowly for all sums to be prime. And the second question gets solved by direct enumeration using again the is_prime R function from the primes package:

[1] 1 1
[1] 1 2
[1] 1 4
[1] 2 3
[1] 3 4

Le Monde puzzle [#1121]

A combinatoric puzzle as Le weekly Monde current mathematical puzzle:

A class of 75<n<100 students is divided at random into two groups of sizes a and b=n-a, respectively, such that the probability that two particular students Ji-ae and Jung-ah have a probability of exactly 1/2 to be in the same group. Find a and n.

(with an original wording mentioning an independent allocation to the group!). Since the probability to be in the same group (under a simple uniform partition distribution) is

it is sufficient to seek by exhaustion values of (a,b) such that this ratio is equal to ½. The only solution within the right range is then (36,45) (up to the symmetric pair). This can be also found by seeking integer solutions to the second degree polynomial equation, namely