Xi'an's Og

A Le Monde mathematical puzzle from after the competition:

A sequence of five integers can only be modified by subtracting an integer N from two neighbours of an entry and adding 2N to the entry.  Given the configuration below, what is the minimal number of steps to reach non-negative entries everywhere? Is this feasible for any configuration?

As I quickly found a solution by hand in four steps, but missed the mathematical principle behind!, I was not very enthusiastic in trying a simulated annealing version by selecting the place to change inversely proportional to its value, but I eventually tried and also obtained the same solution:

      [,1] [,2] [,3] [,4] [,5]
   -3    1    1    1    1
    1   -1    1    1   -1
    0    1    0    1   -1
   -1    1    0    0    1
    1    0    0    0    0

But (update!) Jean-Louis Fouley came up with one step less!

      [,1] [,2] [,3] [,4] [,5]
   -3    1    1    1    1
    3   -2    1    1   -2
    2    0    0    1   -2
    1    0    0    0    0

The second part of the question is more interesting, but again without a clear mathematical lead, I could only attempt a large number of configurations and check whether all admitted “solutions”. So far none failed.

This entry was posted on November 22, 2018 at 12:18 am and is filed under Books, Kids, R with tags CEREMADE, competition, Dauphine, Le Monde, mathematical puzzle, optimisation, R, simulated annealing. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.