Waits Analysis

This chapter is about your chances to win when you are missing exactly one tile to complete legal hand. This is defined as WAIT. COMPLETE set of tables of combinations and matching tiles will be provided.

Small Intro

Neglecting 7 Pairs and Knitted ways to complete a hand we are left with classical 4*Groups + Pair, or 4*3+2 for short (where Group is 3 tiles for Chow or Pung and 4 tiles for Kong).

Some of 3-Groups may be already exposed (you CANNOT exposed a Pair!). Some of concealed Groups may relate to DIFFERENT suits and, hence, have a little space to maneuver. But those CONCEALED-and-in-one-Suit Groups may be re-shuffled in any expected way to form new Groups for winning. We will call these available-for-maneuver tiles as MANEUVER tiles later in the text.

Single-Suited Wait Analysis

We start with analysis in ONE suit neglecting Honors (only two cases H+H and HH+H are available).

All possible cases of those maneuver tiles were identified and classified. There are 9 possible combinations (see table):

Table 1: Single-Suited Waits Distribution

Comments to Table 1:

Some Conclusions

Not surprisingly structure WITHOUT Pair in final hand i.e. 2+1, 5+1, 8+1, 11+1 have very limited calling power (max 3). On the other hand, others provide up to 9-tile-wait (famous Nine Gates 1112345678999). The explanation is simple: a Pair gives flexibility -- some tiles may be used either in 3-Groups or in Pair when adding winning tile (like 444 goes either for Pung or for a Pair + 3rd tile is used for Chow).

Complete Set of Single-Suited Tables

1+1=2              2+1=3              4+1=5              5+1=6              7+1=8              8+1=9              10+1=11          11+1=12            13+1=14

Legend to Single-Suited Tables

First table in each Excel file is showing entire process:

N Tiles Wait Group1 Group2 Group3 Method, where

N = number of waits,

Tiles = tiles in hand

Wait = tiles which legally complete hand

Group... = those pieces which complete hand (Chow, Pung of Pair)

Method = ways how hand is completed, C = Chow, P = Pung, 2 = Pair, Pung are coded prior to Chows, Pair goes last.

 

Please, note, that multiple ways of "breaking" SAME final hand into Groups are considered as one variant.

Example:

66 777 8888 9999 calls for 5, 6 or 7 where after 7 hand could be breaking into:

789*4 + 66, or

678*2 + 789*2 + 99, or

777 + 888 + 999 + 789 + 66

For that hand N=3 though number of lines is 5.

 

Second table simple lists those [unique] variants. In case of last hand there will be one line featuring:

N=3 Tiles=6677788889999.

How many Gates is Nine Gates?

On the top of any cases is 13+1=14 case of Nine Gates with 1112345678999 tiles having 9 waits (see Table 1). But we can find 8 waits for 2 cases in a column 10+1=11 which is also a record. What are they?

2223456777 and 3334567888!!

See, the pattern 311113 calls for 8 tiles, all tiles present in hand PLUS 2 on the SIDE. So, Nine Gates pattern 311111113 SHOULD call for 2 more tiles: 0 and 10 if only those tiles were present in the set! The answer for the question in the title is 9G may be called 11G (if only!).

P.S. Pattern 313 (like 5556777) calls for 5 tiles: 3 from the range and 2 from sides.

Real Table Problem

Click here to see how Single-Suited Waits Analysis works in applications.

Two-Suited Wait Analysis

The simplest case of two-suited wait is XX + YY, getting Pung + Pair. What peculiarities could be found here? Its all about transformation abilities for 3X+2 tiles sets (2, 5, 8 and 11 tiles).

So, Two-Suited waiting hand is simply TWO Chameleons. Waiting tile adds to ONE Chameleon which hence undergoes conversion, the other Chameleons remains untouched.

Complete Set of Chameleon Tables

Chameleons

In this file 1st table is complete list of all chameleons, 2nd table lists unique variants and the 3rd is some kind of summary of Distribution which is provided below in Table 2:

Table 2: Chameleon Waits Distribution

Comments to Table 2:

We read Table 2 in a similar fashion as Table 1. As follows, maximum Number of Waits (for ONE Chameleon) is 3. So, for the waiting hand consisting of 11+2, 8+5 or less tiles maximum Number of Waits in total is 3+2=5.

Examples to Two-Suited Waiting Hands

B11123456 C34555 waiting for B1, B4, B7, C2 and C5 (3+2=5 waits).

B34444567888 C77 waiting for B2, B5, B8 and C7 (3+1=4 waits).

B34555 C55567 waiting for B2, B5, C5 and C8 (2+2=4 waits).

SS RR waiting for S or R (note that Honors were not included into Two-Suited Wait Analysis for the same reason as for Single-Suited).

Next Steps

Next logical step to do is to systemize patterns (see Nine Gates paragraph) of maneuver tiles. Due to 2223444 and, say, 4445666 represent the same animal from the players point of view.



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