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For N≤18, N balls are placed in N consecutive holes. Two players, Alice and Bob, consecutively take two balls at a time provided those balls are in contiguous holes. The loser is left with orphaned balls. What is the values of N such that Bob can win, no matter what is Alice’s strategy?

I solved this puzzle by the following R code that works recursively on N by eliminating all possible adjacent pairs of balls and checking whether or not there is a winning strategy for the other player.

topA=function(awale){
# return 1 if current player can win, 0 otherwise
best=0
if (max(awale[-1]*awale[-N])==1){
#there are adjacent balls remaining
for (i in (1:(N-1))[awale[1:(N-1)]==1]){
if (awale[i+1]==1){
bwale=awale
bwale=0
best=max(best,1-topA(bwale))
}
}}
return(best)
}
for (N in 2:18) print(topA(rep(1,N)))

(brute-force) answering the question that N=5,9,15 are the values where Alice has no winning strategy if Bob plays in an optimal manner. (The case N=5 is obvious as there always remains two adjacent 1′s once Alice removed any adjacent pair. The case N=9 can also be shown to be a lost cause by enumeration of Alice’s options.)