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, 0, 0, -3, 1, -8, 0, -15, 1, -24, 0, -35, 1, -48, 0

Offset: 1

Author

Peter Kagey, Aug 29 2016

Keywords

• sign
• more

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.

Examples

			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]
		

References

• Peter Winkler, Mathematical Puzzles: A Connoisseur's Collection, A K Peters/CRC Press, 2003, pages 1-2.

Crossrefs

Cf. A276163.

Programs

Haskell

minimax [] = 0
minimax as = max (head as - minimax (tail as)) (last as - minimax (init as))
a276168 n = minimum $ map minimax $ permutations [1..n]

Extensions

a(11)-a(16) from Bert Dobbelaere, May 04 2025