"Three Similar Triplets" vs. "Mixed One-Suit" (All Triplets) ============================================================ by Alan Kwan 27 April 2001 Introduction ------------ In Modern Japanese, "Three Similar Triplets" is usually awarded two faan, while "Mixed One-Suit" is awarded three faan if the hand is concealed, or two faan if the hand is exposed. In practice, the frequency of occurence of Three Similar Triplets is very low, without a doubt much lower than that of Mixed One-Suit (and probably even lower than Pure One-Suit). One might suspect that the value of the pattern is too low, that its faan value 'should' be raised. The most common defense against the above accusation is that, since Three Similar Triplets contains three triplets, the hand is "often" an "All Triplets" hand, so the real value of the pattern, namely "Three Similar Triplets + All Triplets", is "often" four faan. Does this make sense? Is four faan really a fair value for "Three Similar Triplets + All Triplets"? Let's verify this by comparing the combination pattern with a similar (and thus easily comparable) one: "Mixed One-Suit + All Triplets". Calculations ------------ Since both patterns are All Triplets hands, it is relatively easy to find their 'combinatorial ratio', by counting the number of possible combinations of triplets and pair in each pattern. Three Similar Triples is easy. There are only 9 choices (9 numbers) for the three similar triplets. The remaining triplet and the pair can be anything. Thus there are: 9 * 31 * 30 = 8370 possible combinations of triplets and pair in Three Similar Triplets + All Triplets hands. Now let's look at Mixed One-Suit. Once we pick a suit, the triplets and pair can be taken from any of the 16 possible tiles. Hence there are: C(16;4,1) = 21840 possible combinations of triplets and pair in Mixed One-Suit + All Triplets hands in a given suit. Since there are three suits, the total possible combinations should be roughly three times that number. There are two complications here. One is that we're counting a few of these hands three times this way: the All Honors hands. The other is that a small minority of these hands are Pure One-Suit hands, which score more than Mixed One-Suit. We can ignore the second issue (as well as the issue that All Honors, too, scores much more than Mixed One-Suit) in our discussion, since doing so could only be unfavorable to the accusation (that the value of Three Similar Triplets is too low). There are C(7;4,1) = 105 combinations for All Honors hands. If we're counting them three times, subtracting twice the count would fix the issue. Thus there are: 21840 * 3 - 105 * 2 = 65310 possible combinations of triplets and pair in Mixed One-Suit + All Triplets hand, in the three suits. Taking ratios, 65310 : 8370 = 7.80... : 1 So Mixed One-Suit + All Triplets hands are 7.8 times as numerous as Three Similar Triplets + All Triplets hands. But while the latter is worth only 4 faan, the former is worth 4 or more faan. We must not forget that unlike Three Similar Triplets, Mixed One-Suit hands do not necessarily contain at least three triplets; there are a lot of Mixed One-Suit hands which are not All Triplet hands, too. Let's think about it: Three Similar Triplets is 7.8 times as rare as Mixed One-Suit among All Triplet hands, and a lot rarer than Mixed One-Suit among non All Triplet hands. Yet the value of the former is not any higher than the value of the latter, in both cases. Conclusion and Thoughts ----------------------- Obviously, the faan value of Three Similar Triplets is too low. The accusation is valid, after all. As we have seen, the defense against the accusation isn't really justifiable; it primarily stems from blind faith, in that the Modern Japanese system, as is, is "ideally and perfectly balanced", which it definitely is not. It is trivial to see that Three Similar Triplets has a combinatorial ratio to Big Three Dragons of 9 to 1. Since the latter is a limit pattern (albeit reportedly the "easiest" one), it's not hard at all to suspect that the former should be worth more than 2 faan. Historically, the value of Three Similar Triplets has remained low in Japanese mahjong, probably because the pattern is so rare that it got 'forgotten' when pattern values were updated. Nowadays, Japanese mahjong players say /sanshoku/ ("three suits") as the abbreviation for "Three Similar Sequences". Won't this be a confusing practice, when there are in fact two /sanshoku/ patterns in the system? The practice has got used and accepted because "Three Similar Triplets" is of hardly any significance within the play strategy concepts of Modern Japanese, because of its rarity coupled with its low value; when someone mentions /sanshoku/, one won't think about "Three Similar Triplets" because one hardly ever thinks about that pattern.