Hong Kong Mahjong · Limit

Thirteen Orphans (十三么) in Hong Kong Mahjong: Limit Hand, 13-Sided Wait and Odds

In Hong Kong mahjong, Thirteen Orphans (十三么) is a limit hand, scored Limit: one each of 1 and 9 characters, 1 and 9 dots, 1 and 9 bamboo, East, South, West, North, Red, Green and White — thirteen kinds — plus a second copy of one of them as the pair. A limit hand scores the 10-fan cap straight away, with nothing added.

It is the only winning hand that doesn't need four sets, which is why so many players look at a hand full of terminals and honours and wonder: should I go for Thirteen Orphans?

The rules for Thirteen Orphans

• All thirteen kinds of terminals and honours, with one of them doubled
• No chows, pungs or kongs — not even a concealed kong: all 14 tiles stay in your hand
• You can win by self-draw or on a discard
• Flowers don't matter: a limit hand is already at the cap, so No Flowers or seat flowers add nothing

The 13-sided wait

If you hold one of each of the thirteen kinds and no pair yet, any of the thirteen kinds wins — the 13-sided wait:

If you already have the pair but one kind is missing, you wait on that kind only. If three or four of it are already in the discards, you are effectively dead, and most players switch to something else.

How Thirteen Orphans is scored

A limit hand isn't added up item by item: once it's Thirteen Orphans, it scores the limit. The fan below are computed live by the same scoring code the game uses.

How it relates to other hands

• Terminals and Honours: also only terminals and honours, but as four pungs — 1 fan plus All Pungs.
• All Terminals: pungs of 1s and 9s with no honours, another limit hand.
• Nine Gates: another concealed limit hand with a many-sided wait, built from 1 to 9 of one suit.
• Heavenly Hand: if the dealer's first 14 tiles are already Thirteen Orphans, that's a Heavenly Hand — also a limit, counted once.

Thirteen Orphans odds

How many kinds of terminals and honours in the opening hand

Thirteen Orphans needs one of each kind, so the number of kinds matters more than the number of tiles. A 13-tile opening hand holds 9 or more kinds about 0.15% of the time, 10 or more 0.01%, and 11 or more only 0.0002%.

Source: 1,000,000 random 13-tile deals (flowers already replaced).

Fourteen random tiles that are already Thirteen Orphans

Counting combinations exactly: 14 random tiles from the 136 (flowers excluded) form Thirteen Orphans with probability 0.000000031% — about once in 3,200,000,000. That's the chance of starting with it; in play you draw and discard, so it shows up far more often.

How often it happens in play

In 100,000 hands of computer play on this site, Thirteen Orphans came up 272 times — once every 368 hands, 0.34% of all wins, 23% of them self-drawn. The computers include Thirteen Orphans when they count how far they are from ready, so with enough terminals and honours they drift into it naturally; the figure reflects tables where someone is willing to try.

Which hands suit Thirteen Orphans

• Many kinds in the opening hand: 9 or more kinds happen only 0.15% of the time, so don't wait for that. With seven or eight kinds and no pairs worth punging, Thirteen Orphans is one road — but a slow one, so watch how fast the others are going.
• Don't pung your honour pairs: one pung ends Thirteen Orphans, so decide early if you hold a scoring honour pair.
• Read the discards: if two or three of a kind you still need are already out, the road narrows fast.
• Others will notice: a stream of discards from 2 to 8 gives it away, and the last terminals you need will be held back.

The biggest cost is speed and no way back: you throw middle tiles all game, hold no sets, and abandoning halfway rarely leaves a usable hand. To try it, open a table of online Hong Kong mahjong and play the computers.

Thirteen Orphans in other mahjong styles

Thirteen Orphans exists in Japanese, Taiwanese and Chinese Official (Guobiao) mahjong, always as a big hand, but the score and the robbing rules vary: Japanese mahjong yaku list, Taiwanese mahjong tai table, Guobiao fan table. Sichuan mahjong has no honour tiles, so it has no Thirteen Orphans.

FAQ

How many fan is Thirteen Orphans?

In Hong Kong mahjong Thirteen Orphans is a limit hand: Limit, the 10-fan cap, with nothing else added.

Can you pung in Thirteen Orphans?

No. All 14 tiles must stay in your hand; any chow, pung or kong — including a concealed kong — ends it.

Can you win Thirteen Orphans on a discard?

Yes. Winning on someone's discard scores the same limit as a self-draw.

What is the 13-sided wait?

When you hold one of each of the thirteen terminals and honours and no pair yet, any of the thirteen kinds completes the hand — the widest wait in mahjong.

Related hands

Terminals & Honours 1 fan Pungs of only 1s, 9s and honours: 1 fan on its own, but it always comes with All Pungs and stacks to the limit fast. All Terminals Limit Four pungs and a pair from just six kinds of 1s and 9s: a limit hand, far harder than Terminals and Honours. Nine Gates Limit 1112345678999 in one suit, waiting on every tile 1 to 9: the ultimate concealed one-suit hand, scored at the limit.