Taiwanese Mahjong · 24 tai

Heavenly Hand (Tian Hu) in Taiwanese Mahjong: Tai, Examples and Odds

In Taiwanese mahjong, the Heavenly Hand (天胡, tian hu) is worth 24 tai: the dealer's 17 opening tiles (after replacing flowers) are already five sets and a pair before a single tile is discarded. It's the biggest single entry in the scoring table.

The rules for the Heavenly Hand

• Only the dealer can get it: the dealer starts with 17 tiles, everyone else with 16
• It counts after the opening flower replacements; replacement tiles are part of the opening hand
• You may not discard anything first, nor declare a concealed kong (a kong ends the first round)
• It's a self-draw, so all three opponents pay

How the Heavenly Hand is scored

The Heavenly Hand replaces self-drawn, concealed hand, concealed self-draw and single wait — those aren't added. Everything else still stacks, such as Pure One Suit, All Pungs, concealed pungs, dragons and flowers. The tai below are computed live by the same scoring code the game uses (dealer and dealer-streak tai are added on top).

Common combinations:

How it relates to other hands

• Concealed self-draw, self-drawn, concealed hand, single wait: replaced by the Heavenly Hand, not added.
• Earthly and Human hands: the same "win in the first round" group — the Heavenly Hand is the dealer's, the other two are the non-dealers'. They never occur together.
• Concealed pungs: every tile was dealt to you, so all pungs are concealed; Three Concealed Pungs and up still count.
• Five Concealed Pungs, All Pungs: five pungs on the deal score the Heavenly Hand, All Pungs and Five Concealed Pungs together.

Heavenly Hand odds

The dealer's 17 tiles are a random draw from the 136 suited and honor tiles (flowers are replaced, so they don't matter). That means the probability can be computed exactly: list every possible count pattern for characters, dots, bamboo and honors, count the ones that split into five sets and a pair, and divide by all possible 17-tile hands.

The result is 0.000084% — roughly once in 1,190,000 deals as dealer.

• That is the chance each time you're dealer; since only the dealer can get it, it's also the chance that anyone at the table gets one in a given hand.
• We don't use the computer-play data for this: it's so rare that tens of thousands of simulated hands would mostly show zero.

What to watch for

• Don't discard yet: as dealer, check whether your 17 tiles already win — once you discard, it's gone.
• Flower replacements count: if the tiles you drew for your opening flowers complete the hand, it's still a Heavenly Hand.
• No kong first: winning on the replacement tile after a concealed kong is Win on a Replacement Tile, not a Heavenly Hand.

To get a feel for being dealer, open a table of online Taiwanese mahjong — the computers play with you and count the tai for you.

The Heavenly Hand in other mahjong styles

• Japanese mahjong: "Tenhou", a yakuman (Japanese mahjong yaku list)
• Hong Kong mahjong: the Heavenly Hand is a limit hand (Hong Kong mahjong fan list)
• Sichuan mahjong: the Heavenly Hand scores the cap (Sichuan mahjong fan list)

FAQ

How many tai is the Heavenly Hand?

In Taiwanese mahjong the Heavenly Hand is worth 24 tai, the biggest single entry in the table. Self-drawn, concealed hand, concealed self-draw and single wait aren't added; everything else stacks, plus dealer tai.

How rare is the Heavenly Hand?

Computed exactly, it's 0.000084% — about once in 1,190,000 deals as dealer.

Can a non-dealer get a Heavenly Hand?

No, only the dealer. A non-dealer self-drawing on their first draw is the Earthly Hand; winning on the dealer's first discard is the Human Hand.

Does it count if I win after replacing opening flowers?

Yes. The 17 tiles after the flower replacements are your opening hand; as long as you haven't discarded or declared a kong, it's a Heavenly Hand.

Related hands

Earthly Hand 16 tai A non-dealer self-draws on their first draw, 16 tai: any chow, pung or kong in the first round cancels it, and the odds are close to a Heavenly Hand's. Human Hand 16 tai A non-dealer wins on the dealer's first discard, 16 tai: you must be ready on the deal, and only the dealer pays. Concealed Self-Draw 3 tai Never claim a tile and self-draw the win — 3 tai. Concealed hand vs self-draw, concealed kongs and replacement wins, and how often it happens.