## Le Monde puzzle [#1075]

**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.

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