## Le Monde puzzle [#977]

**A** mild arithmetic Le Monde mathematical puzzle:

Find the optimal permutation of {1,2,..,15} towards minimising the maximum of sum of all three consecutive numbers, including the sums of the 14th, 15th, and first numbers, as well as the 15th, 1st and 2nd numbers.

**I**f once again opted for a lazy solution, not even considering simulated annealing!,

parme=sample(15) max(apply(matrix(c(parme,parme[-1], parme[1],parme[-(1:2)],parme[1:2]),3),2,sum))

and got the minimal value of 26 for the permutation

14 9 2 15 7 1 11 10 4 12 8 5 13 6 3

Le Monde gave a solution with value 25, though, which is

11 9 7 5 13 8 2 10 14 6 1 12 15 4 3

but there is a genuine mistake in the solution!! This anyway shows that brute force may sometimes fail. (Which sounds like a positive conclusion to failing to find the proper solution! But trying with a simple simulated annealing version did not produce any 25 either…)

October 7, 2016 at 8:14 pm

I found a 25! Using branch&bounding of linear programming instances, similar to an integer optimization algorithm. Since I’m not sure whether spoiling here would be OK, the solution is at http://www.juhokokkala.fi/permutation.txt (just the number sequence, an explanation of the method to be written later.)

October 7, 2016 at 9:41 pm

Thank you, Juho, great news that you found a sequence. (I am adept at spoiling so I would not have minded, but thanks too for the attention.)

October 3, 2016 at 9:02 pm

I hate riddles but when I saw the first version of this post (where neither of the series fitted the announcement – one was too short and the other had two 2’s in it) I actually got going until I quickly had my own sequence with a 26 and then grudgingly decided that other duties were even more urgent…: the reported Le Monde series starts with a 27 – how do I get 25 ? Too lazy for SA myself ;-)

October 3, 2016 at 7:47 am

Hi,

I don’t think the sequences are correct. There are numbers missing.

Cheers,

Anne