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A276168
a(n) is the minimum first-player score difference of a "Coins in a Row" game over all permutations of coins 1..n with both players using a minimax strategy.
1
1, 1, 0, 0, -3, 1, -8, 0, -15, 1
OFFSET
1,5
COMMENTS
A positive value indicates that all permutations of 1..n result in a game where the first player has a winning strategy.
A negative value indicates that there exists a permutation of 1..n where the second player has a winning strategy.
a(2*k) is strictly nonnegative.
Conjecture: a(2*n - 1) = -A067998(n).
Conjecture: a(n) = n + 1 - A276163(n + 1) for all n >= 1.
REFERENCES
Peter Winkler, Mathematical Puzzles: A Connoisseur's Collection, A K Peters/CRC Press, 2003, pages 1-2.
EXAMPLE
a(1) = 1; via [1]
a(2) = 1; via [1,2]
a(3) = 0; via [1,3,2]
a(4) = 0; via [1,2,4,3]
a(5) = -3; via [1,4,2,5,3]
a(6) = 1; via [1,2,3,4,6,5]
a(7) = -8; via [1,5,2,6,3,7,4]
a(8) = 0; via [1,2,3,4,6,5,8,7]
a(9) = -15; via [1,6,2,7,3,8,4,9,5]
PROG
(Haskell)
minimax [] = 0
minimax as = max (head as - minimax (tail as)) (last as - minimax (init as))
a276168 n = minimum $ map minimax $ permutations [1..n]
CROSSREFS
Cf. A276163.
Sequence in context: A096797 A084246 A141252 * A165498 A195731 A331187
KEYWORD
sign,more
AUTHOR
Peter Kagey, Aug 29 2016
STATUS
approved