A394776 Optimal strategy in the game Risk or Safety.

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Offset

1,3

Comments

The game "Risk or Safety" consists of two competing players. A player whose turn it is tosses a coin. If it is heads they earn a point, and they can choose to go for safety, put the point aside in a save box, and give the turn to their opponent. Or they take the risk to continue and toss the coin again. If it is heads again, the number of "open" points increases by one, and they can choose again to continue or stop, converting the "open" points to "saved" points. Whenever it is tails, all the open points are lost, and it is the turn of the other player. Who collects n points first wins.

a(n) denotes the number of times a player should toss a coin, if they still need i points and the opponent needs j points with n = (j-1)^2 + i if i<j and n = i^2 - j + 1 else. A look-up table of a(n) for i and j up to 20 is given in the arXiv paper (see link).

Links

Ruediger Jehn, Table of n, a(n) for n = 1..400

Ruediger Jehn, Optimal strategy in the game Risk or Safety, arXiv:2603.01002 [math.CO], 2026.

Example

a(3) = 2: if both players need 2 more points to win; i.e., i=j=2, the best strategy is to toss twice and to try winning immediately.

Crossrefs

Cf. A387260, A387261, A387262.

Sequence in context: A237615 A343190 A256132 * A340057 A303476 A187201

Adjacent sequences: A394773 A394774 A394775 * A394777 A394778 A394779

Keyword

nonn

Author

Ruediger Jehn, Apr 01 2026

Status

approved