## Le Monde puzzle [#1132]

**A** vaguely arithmetic challenge as Le weekly Monde current mathematical puzzle:

Given two boxes containing x and 2N+1-x balls respectively. If one proceeds by repeatedly transferring half the balls from the even box to the odd box, what is the largest value of N for which the resulting sequence in one of the boxes covers all integers from 1 to 2N?

The run of a brute force R search return 2 as the solution

lm<-function(N) fils=rep(0,2*N) bol=c(1,2*N) while(max(fils)<2){ fils[bol[1]]=fils[bol[1]]+1 bol=bol+ifelse(rep(!bol[1]%%2,2),-bol[1],bol[2])*c(1,-1)/2} return(min(fils))}

with obvious arguments that once the sequence starts cycling all possible numbers have been visited:

> lm(2) [1] 1 > lm(3) [1] 0

While I cannot guess the pattern, there seems to be much larger cases when lm(N) is equal to one, as for instance 173, 174, 173, 473, 774 (and plenty in-between).

April 24, 2020 at 8:49 pm

I just noticed this interesting puzzle. It seems to me that the puzzle asks for the maximum N with the property, and you have found that 2 is the minimum N. I found that 9998 also has the property. Then I stopped looking. I too did not find a pattern to the Ns such that lm(N) = 1. Do you know anything more about the maximum N?

February 27, 2020 at 1:05 am

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