A345383 a(n) is the code of the optimum strategy to win the game 'Super Six' for two players if there are n sticks left in the game.

0, 0, 1, 7, 63, 1023, 32760, 1048544, 33554304, 1073741312, 34359736320, 1099511619584, 35184370221056, 1125899873681408, 36028796752101376

Offset: 1

Author

Ruediger Jehn, Jun 28 2021

Keywords

• nonn
• more

Comments

The rules of Super Six for two players are simple. Game equipment consists of a six-sided die, several 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 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 eliminate all one's sticks before the other player does.

The players take turns. On each turn, the player whose turn it is 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 opposing player.

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 of whether to continue rolling the die or not after successfully placing a stick.

If there are 3 sticks left in the game (i.e., held by a player or standing on the lid), there is just one situation in which a player may have to decide whether to stop or to continue: each player has 1 stick, and 1 stick is on the lid; the best strategy is to continue because the probability of winning is 5 out of 6. Hence, the optimum strategy is "1", where "1" stands for keep rolling the die.

If there are 4 sticks left in the game, there are three situations in which a player may have to make a decision to stop or to continue: in the first, each player has 1 stick, and 2 sticks are on the lid (situation 1); in the second, the player whose turn it is has two sticks, the opposing player has 1 stick, and 1 stick is on the lid (situation 2); in the third, the player whose turn it is has one stick, the opposing player has 2 sticks, and 1 stick is on the lid (situation 3). So there are eight strategies, which may be coded in binary as 000, 001, 010, ... 111, where the digits specify whether the player will continue (1) or stop (0).

The sequence of the situations is defined by 1) the number of sticks on the lid and 2) the number of sticks held by player A (both sorted in descending order). For 5 sticks, the sequence is given by 3/1, 2/2, 2/1, 1/3, 1/2, 1/1, where L/H means there are L sticks on the lid and player A is holding H sticks (necessarily, player B has 5-L-H sticks). Strategy "100" means stopping at situations 1 and 2, and continuing at situation 3.

In the paper "Optimum Strategies for the Game Super Six" (see link below) the situations with 1 stick on the lid, H sticks in the hand of player A and 1 stick in the hand of player B are not considered as situations requiring a strategy. This is because these situations can only occur if a player stops rolling the die when there are zero sticks on the lid. While not a logical event in the strict sense, it is of course a possible situation. - Ruediger Jehn, Oct 05 2021

Examples

			a(4) = 7. With 4 sticks left in the game, there are three situations in which a player has a choice. In all situations, the best strategy is to keep rolling the die. Hence, the best strategy is "111", which, when converted to a decimal number, gives 7.
		

Links

• Rüdiger Jehn, How to win Super Six - Theorem 4, Youtube video, Jun 7 2019.
• Rüdiger Jehn, Optimum Strategies for the Game Super Six, arXiv:2109.10700 [math.GM], 2021.
• Michael Höhle, How to Win a Game (or More) of Super Six, blog, Mar 13 2023.
• Eshed Schacham, Solving Super Six, blog, Aug 2 2024.
• Wikipedia, Super Six (Spiel) (in German)

Crossrefs

Cf. A349697, A349698.

Extensions

a(15) from Ruediger Jehn, Oct 05 2021