A349697 Numerators of the probability that the first player wins the game Super Six if both players have n sticks in their hand and if there are 3 sticks on the lid, assuming optimal play.

78307, 186990749618019112, 1614205257536455860879998130775735700828260230275, 10794889897425456513785608689552167481027910004023676512263195628077085109766959836330846217

Offset: 1

Author

Ruediger Jehn, Kester Habermann and Pontus von Brömssen, Nov 25 2021

Keywords

• nonn
• frac

Comments

The rules of Super Six for two players are as follows. The equipment consists of a six-sided die, a number of sticks, and a box whose lid has six holes. The holes numbered 1 through 5 are shallow, and a stick placed in any one of them will stand up in it; hole #6 goes all the way through the lid so that any stick placed in it falls into the box and is out of play. Initially, an even number of sticks are divided evenly between the two players. The goal is to get rid of all one's sticks before the other player does.

The players take turns. On each turn, the active player rolls the die and places a stick in the numbered hole that matches the number on the die (e.g., a player who rolls a 4 then places a stick in hole #4). The player may roll and place a stick for each roll as many times as desired until rolling a number that is already filled by a stick. When this occurs, the player must take that stick in hand, and play passes to the opponent.

The game proceeds with players taking turns and ends when one player has run out of sticks. The only freedom that the players have is the decision whether to continue rolling the die or not after successfully placing a stick.

The optimum strategy and the winning probabilities can be found in "Optimum Strategies for the Game Super Six" (see link below). The terms of this sequence give the numerators and the terms of sequence A349698 give the denominators of the probability that the first player wins if there are 3 sticks on the lid and both players hold n sticks in their hands, assuming optimal play. If n tends to infinity this probability tends to 1/2.

Examples

			a(1) = 78307 because the probability that the first player wins the game Super Six, when both players have 1 stick and there are 3 sticks on the lid, is 78307/127838 (0.612548...).
		

Links

• Ruediger Jehn, Table of n, a(n) for n = 1..11
• Rüdiger Jehn, Optimum Strategies for the Game Super Six, arXiv:2109.10700 [math.GM], 2021.
• Wikipedia, Super Six (in German)

Crossrefs

Cf. A345383, A349698.