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세미나

세미나

원드라이브

Mathematical analysis of Mahjong

tile

Definition

Let non-empty finite set $\mathcal{C}$ be a color set and $\mathcal{N} \subset \mathbb{N}$ be a suit set. A tile $x_{k} \in \mathcal{C} \times \mathcal{N}$ is defined by a pair of color and integer.

Example

For ordinary mahjong, $\mathcal{C} = \left\{ b , c , d \right\}$ is consists three color. $b$, $c$, $d$ stand for bamboo, character, dot, respectively. More commonly, $\mathcal{C}$ also has wind $w$ and dragon $\delta$.

  • $b_{3}$-$b_{6}$ are mutually suzi, unlike $b_{3}$-$b_{4}$ or $b_{3}$-$c_{3}$.
  • Usually, $x_{k}$ for $k > 9$ doesn’t exist.
  • We say wind tile and dragon tile as honor. The honor tile has no suit, so subscript may refer specific identifier, like north wind $w_{N}$ and green $\delta_{G}$.
  • Non-terminal and non-honor tile is called by simple.

taz

Definition

A multiset of $n$ tiles $X = \left\{ x_{1} , \cdots , x_{n} \right\}$ is called by $n$-taz. The suit range of taz $X$, $R(X)$ is defined by difference of maximal suit and minimal suit of all simple tile in $X$.

Example

  • A $14$-taz is called by hand.
  • A $2$-taz with two same tiles is called by toiz.
  • A $3$-taz is called by menz or meld.
    • A menz with all same tiles is called by kutz.
    • A menz with serial tiles is called by shunz, $\left\{ X_{k-1} , X_{k} , X_{k+1} \right\}$.
  • Here is called “Nine Gates” hand: $$ \left\{ d_{1}, d_{1}, d_{1}, d_{2}, d_{3}, d_{4}, d_{5}, d_{6}, d_{7}, d_{8}, d_{9}, d_{9}, d_{9} \right\} $$

wait and better taz

The wait of $n$-taz $X$, $W(X)$ is a number of $(n+1)$-taz including $X$ and the difference of suit range is less than 3. That is, $$ W ( X ) = \operatorname{card} \left\{ Y \supset X : |Y| = n + 1 \land R(Y) < R(X) + 3 \right\} . $$ For two $n$-tazs $A$ and $B$, we say $A$ is better than $B$ and denote $A > B$ if $W(A) > W(B)$. Similiary, the equality $A = B$ holds if $W(A) = W(B)$.

Theorem

The terminal tile is the worst. $$ \left\{ x_{k} \right\} > \left\{ x_{1} \right\} = \left\{ x_{9} \right\} \qquad , \forall k \in [2, 8] $$

proof

Without loss of generality, we only have to observe $x_{1}, x_{2}, x_{3}$.

There are only three $2$-taz including $x_{1}$: $$ \left\{ x_{1} , x_{1} \right\} , \left\{ x_{1} , x_{2} \right\} , \left\{ x_{1} , x_{3} \right\} $$

There are four $2$-taz including $x_{2}$: $$ \left\{ x_{1} , x_{2} \right\} , \left\{ x_{2} , x_{2} \right\} , \left\{ x_{2} , x_{3} \right\} , \left\{ x_{2} , x_{4} \right\} $$

There are five $2$-taz including $x_{3}$: $$ \left\{ x_{1} , x_{3} \right\} , \left\{ x_{2} , x_{3} \right\}, \left\{ x_{3} , x_{3} \right\} , \left\{ x_{3} , x_{4} \right\} , \left\{ x_{3} , x_{5} \right\} $$

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