## Le Monde puzzle [#910] An game-theoretic Le Monde mathematical puzzle:

A two-person game consists in choosing an integer N and for each player to successively pick a number in {1,…,N} under the constraint that a player cannot pick a number next to a number this player has already picked. Is there a winning strategy for either player and for all values of N?

for which I simply coded a recursive optimal strategy function:

```gain=function(mine,yours,none){
fine=none
if (length(mine)>0)
fine=none[apply(abs(outer(mine,none,"-")),
2,min)>1]
if (length(fine)>0){
rwrd=0
for (i in 1:length(fine))
rwrd=max(rwrd,1-gain(yours,c(mine,fine[i]),
none[none!=fine[i]]))
return(rwrd)}
return(0)}
```

which returned a zero gain, hence no winning strategy for all values of N except 1.

```> gain(NULL,NULL,1)
 1
> gain(NULL,NULL,1:2)
 0
> gain(NULL,NULL,1:3)
 0
> gain(NULL,NULL,1:4)
 0
```

Meaning that the starting player is always the loser!

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