"Thirteen Orphans" vs. "Nine Gates of Heaven"
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by Alan Kwan 6 September 2000
Introduction
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Does "Thirteen Orphans" deserve to be crowned "the king of
mahjong hands"? Is it really the most difficult mahjong hand?
This article tries to use elementary combinatorics to compare
Thirteen Orphans (Thirteen Terminals) against Nine Gates of
Heaven (Sacred Lamp of Nine Lotus, Nine Connected Pieces), the
mahjong hand which some hold to be the most "perfect". The
reader is assumed to possess some background in elementary
combinatorics.
Here, we use the original, Chinese Classical definition for Nine
Gates, which requires that the hand is actually calling for 9
tiles before it goes out. (The looser Modern Japanese definition
allows any hand which includes the specified shape, even if one
of the tiles in the shape is picked up as the hand goes out.)
Because this hand is defined on the 13-tile calling hand instead
of the 14-tile winning hand, in order to keep the calculations
simple, we will be comparing the 13-tile calling hand of Thirteen
Terminals with this calling hand. We'll leave the interpretation
of the result to the reader, who should take into account the
fact that these two calling hands have different chances of going
out.
Calculations
------------
We do the same thing we've been doing before: we simplify the
problem by comparing the number of 13-tile hands in the
136-tile set which are calling hands of each pattern in question.
(What I call 'finding the "combinatorial ratio"'.)
How many 13-tile hands are calling hands of Thirteen Orphans?
There are two types of such hands: the 13-way call hand with 13
different terminals, and the 1-way call hand with 1 pair and 11
different ones. There are
C(4,1)^13
= 4^13
hands of the former, and
C(13;11,1) * C(4,1)^11 * C(4,2)
= 4^11 * 936
hands of the latter.
Adding these together, there are
4^13 + 4^11 * 936
= 4^11 * (16 + 936)
= 4^11 * 952
13-tiles hand in the 136-tile set which are calling hands of
Thirteen Orphans.
Now, how many hands are calling hands of Nine Gates? There are
three suits, and the hand must be of the shape 1112345678999.
There are a total of
3 * C(4,3)^2 * C(4,1)^7
= 3 * 4^9
such hands in the set.
Taking ratios,
4^11 * 952 : 3 * 4^9
= 16 * 952 : 3
= 5077.33... : 1
or roughly 5000 to 1 ! Nine Gates calling hands are 5000 times
as 'rare' as Thirteen Orphans calling hands in the 136-tile set!
Interpretation & Conclusion
---------------------------
We should not forget that the Nine Gates calling hand is probably
much easier to go out. It's calling for 23 tiles out of the
remaining 123, while the vast majority of the Thirteen Orphans
calling hands are calling for only 4 tiles. But even if we are
generous and take it to be a 1:6 ratio, this is clearly
overwhelmed by the 5000 to 1 combinatorial ratio between the
13-tile calling hands.
Note that the Combinatoric ratio does not translate to a ratio of
the practical frequency of the hands. Because of the
draw-and-discard play mechanism, Thirteen Orphan calling hands
occur perhaps only tens or hundreds of times as often as Nine
Gates calling hands. But in any case, we can safely conclude
that Nine Gates is a lot rarer and harder than Thirteen Orphans.
In practice, if one is dealt a hand with many different
terminals, one often doesn't have much choice other than
attempting Thirteen Orphans. But if one is dealt a good Pure
One-Suit hand, one would often prefer to go out with Pure
One-Suit instead of risking it to go for Nine Gates.
For this reason, we can expect an even lower frequency for the
completion of Nine Gates in practice.
Before we close the discussion, let's consider the looser, Modern
Japanese definition of Nine Gates. I'm omitting the calculations
here: that pattern (the 14-tile completed hand) has a
'combinatorial ratio' to Thirteen Orphans of roughly 1 to 151.
Even the looser Modern Japanese definition of Nine Gates is a lot
harder than Thirteen Orphans.