Hi. Here's another article of mine which no one can understand. ^_^
This article can be downloaded from: http://home.netvigator.com/~tarot/Mahjong/mj_com05.txt
If you find any errors, please kindly point them out.
by Alan Kwan 6 September 2000
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.
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!
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 probably occur 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 sacrificing it for the sake of 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 454. Even the looser Modern Japanese definition of Nine Gates is a lot harder than Thirteen Orphans.
"Live life with Heart." - Alan Kwan / tarot@notmenetvigator.com
http://home.netvigator.com/~tarot (hard-core video game reviews)
Tarot Games Hong Kong: http://home.netvigator.com/~tarot/com
(please remove anti-spam section "notme" from mailing address)