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**A** Le Monde mathematical puzzle in combinatorics:

Given a permutation σ of {1,…,5}, ifσ(1)=n, the n first values of σ are inverted. If the process is iterated until σ(1)=1, does this always happen and if so what is the maximal number of iterations? Solve the same question for the set {1,…,2014}.

**I** ran the following basic R code:

N=5 tm=0 for (i in 1:10^6){ sig=sample(1:N) #random permutation t=0;while (sig[1]>1){ n=sig[1] sig[1:n]=sig[n:1] t=t+1} tm=max(t,tm)}

obtaining 7 as the outcome. Here is the evolution of the maximum as a function of the number of terms in the set. If we push the regression to N=2014, the predicted value is around 600,000. .. Running a million simulations of the above only gets me to 23,871!A wee reflection lead me to conjecture that the maximum number of steps w_{N} should be satisfy w_{N}=w_{N-1}+N-2. However, the values resulting from the simulations do not grow as fast. Monte Carlo effect or true discrepancy?

Filed under: Books, Kids, R Tagged: Le Monde, mathematical puzzle, R

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