"Mixed Terminals 7 Pairs" vs. "13 Orphans"
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by Alan Kwan  15 April 1999

「混老七対」ｖｓ．「国士無双」
作者：関兆豪

(Note:  This discussion pertains to Modern Japanese, though
the results may be applicable to other styles.)

Is "7 Pairs" combined with "Mixed Terminals" (i.e. 7 pairs of
terminals and honors) really harder than "13 Orphans"?  Let's use the
tool of Combinatorics to try to find some insight.

「混老七対」と「国士」、どちらが難しいか、数学的に数えて
みましょう。

Since both hands are irregular hands which cannot make use of claimed
discards (before going out), a straight comparison of their
combinatoric multiplicity is a very good pointer (though not the
entire picture).

両方ともチー、ポンを利用できない特殊な手なので、数えて
その組合せの数を素直に比べるのは、かなり良い指標になります。

We'll count the number of 14-tile combinations taken from the 136-tile
deck which form the respective hands.  The counting is rather simple,
and only basic combinatorics are involved.  For the sake of brevity,
"terminal" means "terminal or honor" for the remainder of the article.

全１３６枚の牌から１４枚の組合せを選んで、それぞれの手役
が出来る組合せの数を数えます。以下、「幺九牌と字牌」を
「幺九牌」と略記します。

Let's start with "13 Orphans".  The hand consists of 1 pair of
terminals, and 1 of each of the remaining 12 terminals.  Just a
standard selection exercise:

国士は１３種類の幺九牌から１つを選んで、その４枚から２枚を
取って雀頭にします。そして、残りの１２種類は１枚ずつ。
分る人には簡単でしょう：

C(13,1) * C(4,2) * C(4,1)^12
= 13 * 6 * 4^12
= 1 308 622 848

Now let's look at "7 Pairs Mixed Terminals".  Simply, 7 pairs of
terminals.  Another simple exercise:

混老七対は１３種類の幺九牌から７種類を選んで、それぞれ
２枚ずつ：

C(13,7) * C(4,2)^7
= 1716 * 6^7
= 480 370 176

The first quantity is roughly 2.72 times that of the second.  This
means that we can say that "7 Pairs Mixed Terminals" is somewhat
harder than "13 Orphans".

前者は後者の約２．７２倍もあります。つまり、「混老七対」は
「国士」より「多少難しい」と言っていいでしょう。

In practice, the large discrepency between the two pattern values
further discourages attempts at "7 Pairs Mixed Terminals"; often one
would give up and settle for just "7 Pairs".  Thus we can expect "7
Pairs Mixed Terminals" to be completed a lot less frequently than "13
Orphans" in practice.

実際には、「混老七対」の点がかえって大部低いので、プレイヤー
はあまり執着に狙わなくて、「混老」を諦めて「ただの七対」などに
走ることは多いなので、実際に手役が出来る頻度は、「国士」より
「かなり低い」と予想できます。