## Le Monde puzzle [#810]

**T**he current puzzle is as follows:

Take a board with seven holes and seeds. The game starts with one player putting the seeds on the holes as he or she wishes. The other player picks a seed wherever. Then, alternatively, each player picks a seed in a hole contiguous to the previous one. The loser is the one finding only empty holes to pick from. Who is the winner with 28? 29? 30 seeds?

**T**his is a simplified version of the awalĂ© or oware we used to play with my kids.

**I** first defined a recursive function on the win/loose value of a particular location, based on the assumption that each player was picking the best location at each step:

f=function(x,i){ if (x[i]==0){# losing location v=0;return(v)}else{ if ((i>1)&&(i<7)){ x[i]=x[i]-1;return(1-max(f(x,i-1),f(x,i+1))) }else{ if (i==1){ x[i]=x[i]-1;return(1-f(x,2))} if (i==7){ x[i]=x[i]-1;return(1-f(x,6))} } }}

and then checked whether or not winning solutions were available for 28, 29, and 30 seeds dropped at random:

N=28 #number of seeds glosol=1 #boolean for (t in 1:10^3){#random starts seeds=sample(1:7,N,rep=TRUE) x=rep(0,7) for (i in 1:7) x[i]=sum(seeds==i) sol=i=0 #second player result while ((i<7)&&(sol==0)){ i=i+1;sol=f(x,i)} if (sol==0){ #winning configuration for first player glosol=0;print(x);break()} }

getting solutions for 28 (5 6 2 3 5 5 2) and 30 (6 6 2 4 4 5 3), but none for 29.

**A**ctually, the rule seems to be that odd numbers get no solutions and even numbers get solutions (e.g., 1 1 1 1 1 2 1 for 8 seeds). (This means further that to build a winning allocation for 2N seeds, we only need to take a configuration at random with 2N+1 seeds and check which seed we need to remove to get a winning (for the “other” player) configuration.)

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