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(This article was first published on <strong><a href="https://xianblog.wordpress.com/2018/11/22/le-monde-puzzle-1075/"> R – Xi'an's Og</a></strong>, and kindly contributed to <a rel="nofollow" href="https://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><strong><a rel="nofollow" href="https://xianblog.files.wordpress.com/2015/10/lemondapari.jpeg" target="_blank"><img src="https://xianblog.files.wordpress.com/2015/10/lemondapari.jpeg?w=450&h=299&fit=456%2C456" data-recalc-dims="1"></a>A</strong> <a rel="nofollow" href="https://xianblog.wordpress.com/2011/09/03/le-monde-puzzle-website/" title="Le Monde puzzle [website]" target="_blank">Le Monde mathematical puzzle</a> from after the competition:</p>
<blockquote>
<p><em>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?<a rel="nofollow" href="https://xianblog.files.wordpress.com/2018/11/lemonde1075.jpg" target="_blank"><img data-orig-file="https://xianblog.files.wordpress.com/2018/11/lemonde1075.jpg?w=202&amp;h=194&amp;fit=202%2C194&amp;resize=202%2C194" data-orig-size="343,329" data-medium-file="https://xianblog.files.wordpress.com/2018/11/lemonde1075.jpg?w=202&amp;h=194&amp;fit=202%2C194&amp;resize=202%2C194?w=300" data-attachment-id="37513" data-image-meta="{" data-image-description="" src="https://xianblog.files.wordpress.com/2018/11/lemonde1075.jpg?w=202&h=194&fit=202%2C194&resize=202%2C194" data-permalink="https://xianblog.wordpress.com/2018/11/22/le-monde-puzzle-1075/lemonde1075/" data-large-file="https://xianblog.files.wordpress.com/2018/11/lemonde1075.jpg?w=202&amp;h=194&amp;fit=202%2C194&amp;resize=202%2C194?w=343" alt="" data-image-title="lemonde1075" sizes="(max-width: 202px) 100vw, 202px" width="202" data-comments-opened="1" class="wp-image-37513" data-recalc-dims="1" height="194"></a><br>
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<p>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 <a rel="nofollow" href="http://www.ceremade.dauphine.fr/~xian/lemonde1074.R" target="_blank">eventually tried</a> and also obtained the same solution:</p>
<pre>      [,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
</pre>
<p>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 <a rel="nofollow" href="http://www.ceremade.dauphine.fr/~xian/lemonde1074.R" target="_blank">none</a> failed.</p>
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