## IMPORTANT

**Conny Hansson has pointed out a potential flaw in the numbers below. This chart correctly counts the number of ways to make a scoring pattern, but it does not account for arranging the tiles. This means the total number of regular hands is larger than the combinations of tiles that would make those hands.

To illustrate: Imagine three chows of 123 dots, plus a pung of 7 dots, plus a pair of 9 dots. Those same tiles could make pungs of 1, 2, 3, and 7 of dots with a pair of 9's. This chart counts those hands separately. Whether it should or not is another matter.

This table can be used to compare the relative frequency of obtaining each of the scoring elements in a random collection of tiles that happen to be a regular hand.

It shows the number of possible ways to create each of the scoring patterns. That is to say the number of combinations of tiles in all possible regular hands. No regard is given to game play: no chow, pung, or kong claims are considered. In practice, this means that patterns with sequences will have a higher result than triplets (as there are more tiles available), and that multiple honor sets will be easier to obtain than similar numbered sets (as players discard single honors more frequently).

Nothing that requires a wait (such as Nine Gates, Edge Wait, etc.) are possible to compute with the same methods, nor is any computation involving having concealed versus exposed sets.

**Note** that Regular Hands means hands that have four sets of three tiles and a pair. The table also has irregular hands in it (such as Seven Pairs), but they are **not** included in the total count for regular hands. Similarly, hands with irregular elements (such as anything knitted) is also **not** counted in that total.

# Table Key

Column Name | Meaning |
---|---|

A | World Series Of Mahjong Overview (aka Zung Jung) v3.2 Fan Number |

B | Zung Jung v3.2 Score |

C | Chinese Official Overview (aka Mahjong Competition Rules) Fan Number |

D | MCR Score |

E | World Mahjong Federation Overview (aka WMPA) Fan Number |

F | WMFed Score |

G | EMA Riichi Yaku - Concealed(Exposed) - * Special |

# Results

Scoring Pattern | Combinations | A | B | C | D | E | F | G |
---|---|---|---|---|---|---|---|---|

Regular Hands | 11,369,205,075,492 | |||||||

No Honors | 7,154,156,054,616 | 76 | 1 | |||||

Four Chows | 6,715,895,624,442 | 1.1 | 5 | |||||

Four Chows No Value Honors | 5,339,360,797,812 | *1(0) | ||||||

Mixed Double Chow | 4,973,054,207,424 | 70 | 1 | |||||

Four Chows No Honors | 4,788,746,867,160 | 63 | 2 | 15 | 1 | |||

One Pung | 3,809,325,606,336 | |||||||

Short Straight | 3,537,856,860,870 | 71 | 1 | |||||

One Voided Suit | 2,633,031,836,760 | 75 | 1 | |||||

Pung of Terminal or Wind | 1,828,674,922,116 | 73 | 1 | |||||

Value Pair | 1,786,219,355,376 | |||||||

Seven Pairs | 1,569,298,171,584 | 10.2 | 30 | 19 | 24 | |||

Seven Unique Pairs | 1,505,948,184,576 | 24 | 2 | 2 | ||||

Two Suits Only | 1,158,898,125,528 | |||||||

Two Terminal Chows | 978,440,223,168 | 72 | 1 | |||||

All Simples | 903,760,799,976 | 1.3 | 5 | 68 | 2 | 13 | 1 | 1 |

Mixed Shifted Chows | 823,042,473,984 | 51 | 6 | |||||

Value Honor | 820,346,243,136 | 3.1 | 10 | |||||

Two Pungs | 774,405,495,000 | |||||||

Dragon Pung | 564,711,862,608 | 59 | 2 | 6,7,8 | 1 | 1 | ||

Two Identical Chows | 448,675,374,720 | 5.1.1 | 10 | 69 | 1 | 14 | 1 | 1(0) |

Mixed Triple Chow | 357,580,366,530 | 6.1 | 35 | 48 | 8 | 19 | 2 | 2(1) |

Two Similar Chows Twice | 267,760,375,104 | |||||||

Knitted Straight | 237,892,534,272 | 35 | 12 | |||||

Seat Wind | 208,720,839,984 | 61 | 2 | 4 | 1 | 1 | ||

Prevalent Wind | 208,720,839,984 | 60 | 2 | 5 | 1 | 1 | ||

Knitted Straight, Set and Pair | 204,069,666,816 | 35 | 12 | |||||

Mixed Straight | 203,069,423,616 | 39 | 8 | |||||

Lesser Honors and Knitted Tiles | 135,291,469,824 | 34 | 12 | |||||

Straight | 116,688,273,408 | 7.1 | 40 | 28 | 16 | 18 | 2 | 2(1) |

Tile Hog | 108,496,998,180 | 64 | 2 | |||||

Six Honors and Knitted Tiles | 101,468,602,368 | 34 | 12 | |||||

Small Two Winds | 92,739,602,304 | |||||||

All Types | 72,593,123,328 | 52 | 6 | 17 | 2 | |||

Three Pungs | 67,441,343,634 | |||||||

Three Pure Shifted Chows | 58,253,661,696 | 30 | 16 | |||||

Greater Honors and Knitted Tiles | 57,982,058,496 | 20 | 24 | |||||

Two Consecutive Pungs | 42,449,544,000 | |||||||

Double Pung | 41,281,971,264 | 65 | 2 | |||||

Five Honors and Knitted Straight | 33,822,867,456 | 34+35 | 24 | |||||

All Fives | 29,819,607,684 | 31 | 16 | |||||

Outside Hand | 28,400,308,860 | 8.1.1 | 40 | 55 | 4 | 16 | 2 | 2(1) |

Half Flush | 21,438,061,428 | 2.1.1 | 40 | 50 | 6 | 26 | 3 | 3(2) |

Big Two Winds | 13,352,952,960 | |||||||

Big Two Dragons | 6,986,887,488 | 54 | 6 | |||||

Terminal in Each Set | 6,368,641,140 | 8.1.2 | 50 | 28 | 3 | 2 | ||

Upper Four | 2,938,382,904 | 36,37 | 12 | |||||

All Pungs | 2,137,006,080 | 4.1 | 30 | 49 | 6 | 23 | 2 | 2 |

Full Flush | 1,446,713,304 | 2.1.2 | 80 | 22 | 24 | 31 | 6 | 6(5) |

Two Identical Chows Twice | 1,399,133,304 | 5.1.2 | 60 | 27 | 3 | 3(0) | ||

Thirteen Orphans | 1,308,622,848 | 10.1 | 160 | 7 | 88 | 40 | 15 | *L |

Small Three Similar Pungs | 1,253,507,136 | 6.2.1 | 30 | |||||

Small Three Winds | 1,055,296,512 | 3.3.1 | 30 | |||||

Reversible Tiles | 931,561,760 | 40 | 8 | |||||

Mixed Shifted Pungs | 459,835,392 | 42 | 8 | |||||

Four Pure Shifted Chows | 434,221,056 | 16 | 32 | |||||

Three-Suited Terminal Chows | 301,989,888 | 29 | 16 | |||||

Three Identical Chows | 279,401,280 | 5.1.3 | 120 | 23 | 24 | |||

Three Pure Shifted Pungs | 278,299,266 | 7.2.1 | 100 | 24 | 24 | |||

Small Three Dragons | 265,510,656 | 3.2.1 | 40 | 10 | 64 | 30 | 4 | 2 |

Three Similar Pungs | 103,947,264 | 6.2.2 | 120 | 32 | 16 | 20 | 2 | 2 |

Upper Tiles | 63,943,416 | 25,26,27 | 24 | |||||

Big Three Winds | 62,115,840 | 3.3.2 | 120 | 38 | 12 | |||

Big Three Dragons | 16,137,216 | 3.2.2 | 130 | 2 | 88 | 33 | 12 | L |

All Terminals and Honors | 9,838,080 | 8.1.3 | 100 | 18 | 32 | 29 | 4 | 2 |

Two Similar Pungs Twice | 7,741,440 | |||||||

All Even | 6,082,560 | 21 | 24 | |||||

Seven Shifted Pairs | 2,519,424 | 6 | 88 | |||||

Small Four Winds | 2,248,704 | 3.3.3 | 320 | 9 | 64 | 32 | 12 | L |

Pure Terminal Chows | 839,808 | 13 | 64 | |||||

Four Consecutive Pungs | 829,440 | 7.2.2 | 200 | 15 | 48 | |||

All Honors | 161,280 | 3.4 | 320 | 11 | 64 | 37 | 12 | L |

All Green | 155,316 | 3 | 88 | L | ||||

Big Four Winds | 46,080 | 3.3.4 | 400 | 1 | 88 | 41 | 20 | 2L |

All Terminals | 46,080 | 8.1.4 | 400 | 8 | 64 | 39 | 15 | L |

Four Identical Chows | 3,906 | 5.1.4 | 480 | 14 | 48 |

# Methodology

For the regular hands (four sets and a pair), the first set steps through all of the pungs, then all of the chows. Each of the other three sets steps from the previous set to the end of that list (so as to avoid counting permutations). The pair then steps through each of the 34 tile types. (So the first combination examined is four pungs of one dot, and a pair of the same, which would require 14 identical tiles!) Any combinations that require more than four of any one tile are skipped. (So the first combination counted has pungs of 1 dot, 2 dots, 3 dots, 4 dots, and a pair of 5 dots.)

A count of ways to combine tiles to make that hand is then counted. (Continuing the example, there are 4 ways to make each of the dot pungs, and 6 ways to make the pair, or 4 * 4 * 4 * 4 * 6 or 1536 ways to make that hand.) Then each of the patterns is tested to see if the hand conforms, testing the more-restrictive patterns first, and the count is added to the pattern's total if the test passes. (The example counts towards Four Consecutive Pungs, but not Three Consecutive Pungs.) Further implications are not removed. (Our example hand is counted in All Pungs.)

For seven pairs hands, each of the seven pairs is stepped from the previous set through the remaining tiles.

Some of the results are computed directly such as Thirteen Orphans is (4^12 * 6 * 13). Greater Honors and Knitted Tiles as well as Lesser Honors and Knitted Tiles are computed similarly. Someone please check my math!

The list of exclusions:

- Upper Four excludes Upper Tiles (That is, a hand that scores for Upper Tiles is NOT counted toward Upper Four)
- One Pung excludes two or more; Two Pungs excludes three or more; Three Pungs excludes four pungs
- Two Terminal Chows excludes Three Suited- and Pure-Terminal Chows
- Outside Hand excludes Pure Lesser Terminals, Terminals and Honors, All Terminals
- Pure Lesser Terminals excludes All Terminals
- All Terminals excludes All Terminals and Honors
- Half- and Full-Flush exclude All Honors
- Half Flush excludes Full Flush
- Big Four Winds excludes all smaller Winds; etc.
- Big Three Dragons excludes all smaller Dragons; etc.
- Mixed Double Chow excludes Two Similar Sequences Twice
- Two Identical Sequences excludes three or more identical sequences; etc.
- Short Straight excludes Straight
- Three Pure Shifted Chows excludes Four Pure Shifted Chows
- Double Pung excludes Double Pung Twice
- Two Consecutive Pungs excludes three or more; etc.

If you have any questions about the table, please ask away. Also, feel free to suggest improvements, or just add them yourself.