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.)

• Players alternate moves.
• A move consists of exchanging a card on the table with a lower card from the other pile.
• The player whose move makes the sum of the cards on the table under 21 loses.

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

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

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  

• Initial deal: 0,2,4,6,7,11. The total is $30$. The pattern for this deal is

  $\bullet$		$\bullet$
  	$\bullet$	$\bullet$
  $\bullet$			$\bullet$

  where $\bullet$ is a $\pm$. No combinations of $\pm$ choices will yield a tetracode odd men out, so this deal is not a hexad.
• First player replaces 7 by 5: 0,2,4,5,6,11. The total is now 28. (Note this is a square in the picture at 1.) This corresponds to the col+tet

  +		+
  +	+
  -			+

  with tetracode odd men out $-\ + \ 0\ -$.
• Second player replaces 11 by 7: 0,2,4,5,6,7. The total is now 24. Interestingly, this $6$-set corresponds to the pattern

  $\bullet$		$\bullet$
  $\bullet$	$\bullet$	$\bullet$
  $\bullet$

  (hence possibly with an allowed odd men out $0\ + \ +\ ?$, for example). However, it has column distribution $3,1,2,0$, so it cannot be a hexad.
• First player replaces 6 by 3: 0,2,3,4,5,7. (Note this is a cross in the picture at 0.) This corresponds to the tet-tet pattern

  	+	-
  +	-	+
  -

  with tetracode odd men out $-\ - \ +\ 0$. Cards total 21. First player wins.