## Le Monde puzzle [#1110]

**A** low-key sorting problem as Le Monde current mathematical puzzle:

If the numbers from 1 to 67 are randomly permuted and if the sorting algorithm consists in picking a number i with a position higher than its rank i and moving it at the correct i-th position, what is the maximal number of steps to sort this set of integers when the selected integer is chosen optimaly?

As the question does not clearly state what happens to the number j that stood in the i-th position, I made the assumption that the entire sequence from position i to position n is moved one position upwards (rather than having i and j exchanged). In which case my intuition was that moving the smallest moveable number was optimal, resulting in the following R code

sorite<-function(permu){ n=length(permu) p=0 while(max(abs(permu-(1:n)))>0){ j=min(permu[permu<(1:n)]) p=p+1 permu=unique(c(permu[(1:n)<j],j,permu[j:n]))} return(p)}

which takes at most n-1 steps to reorder the sequence. I however checked this insertion sort was indeed the case through a recursive function

resorite<-function(permu){ n=length(permu);p=0 while(max(abs(permu-(1:n)))>0){ j=cand=permu[permu<(1:n)] if (length(cand)==1){ p=p+1 permu=unique(c(permu[(1:n)<j],j,permu[j:n])) }else{ sol=n^3 for (i in cand){ qermu=unique(c(permu[(1:n)<i],i,permu[i:n])) rol=resorite(qermu) if (rol<sol)sol=rol} p=p+1+sol;break()}} return(p)}

which did confirm my intuition.

September 17, 2019 at 8:11 am

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