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Mathematical blackjack

Mathematical blackjack is a 2-person combinatorial game whose rules will be described below. What is remarkable about it is that a winning strategy, discovered by Conway and Ryba [CS2] and [KR], depends on knowing how to determine hexads in the Steiner system $S(5,6,12)$ using the shuffle labeling.
Winning ways in mathematical blackjack
Mathematical blackjack is played with 12 cards, labeled $0,...,11$ (for example: king, ace, $2$, $3$, ..., $10$, jack, where the king is $0$ and the jack is $11$). Divide the 12 cards into two piles of $6$ (to be fair, this should be done randomly). Each of the $6$ cards of one of these piles are to be placed face up on the table. The remaining cards are in a stack which is shared and visible to both players. If the sum of the cards face up on the table is less than 21 then no legal move is possible so you must shuffle the cards and deal a new game. (Conway [Co2] calls such a game *={0|0}, where 0={|}; in this game the first player automatically wins.) The winning strategy (given below) for this game is due to Conway and Ryba [CS2], [KR]. There is a Steiner system $S(5,6,12)$ of hexads in the set $\{0,1,...,11\}$. This Steiner system is associated to the MINIMOG of in the "shuffle numbering" rather than the ``modulo $11$ labeling''.

Proposition 6 (Ryba)   For this Steiner system, the winning strategy is to choose a move which is a hexad from this system.

This result is proven in [KR]. If you are unfortunate enough to be the first player starting with a hexad from $S(5,6,12)$ then, according to this strategy and properties of Steiner systems, there is no winning move. In a randomly dealt game there is a probability of

\begin{displaymath} {132\over{\left( \begin{array}{c} 12 \\ 6\end{array}\right)}} =1/7 \end{displaymath}

that the first player will be dealt such a hexad, hence a losing position. In other words, we have the following result.

Lemma 7   The probability that the first player has a win in mathematical blackjack (with a random initial deal) is $6/7$.

Example 8  


Next: Bibliography Up: MINIMOGs and Mathematical blackjack Previous: The shuffle kitten   Contents
David Joyner
2000-05-29