A394776 - OEIS

%I #12 Apr 10 2026 01:03:42

%S 1,1,2,2,1,2,1,3,3,1,1,1,2,2,2,4,1,1,1,1,2,2,2,2,5,1,1,1,1,1,2,2,2,2,

%T 2,6,1,1,1,1,1,1,2,2,2,2,2,3,7,1,1,1,1,1,1,1,2,2,2,2,2,2,3,8,1,1,1,1,

%U 1,1,1,1,2,2,2,2,2,2,3,3,9,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,10,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,11

%N Optimal strategy in the game Risk or Safety.

%C 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.

%C 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).

%H Ruediger Jehn, <a href="/A394776/b394776.txt">Table of n, a(n) for n = 1..400</a>

%H Ruediger Jehn, <a href="https://arxiv.org/abs/2603.01002">Optimal strategy in the game Risk or Safety</a>, arXiv:2603.01002 [math.CO], 2026.

%e 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.

%Y Cf. A387260, A387261, A387262.

%K nonn

%O 1,3

%A _Ruediger Jehn_, Apr 01 2026