Hong Kong Mahjong

A Hong Kong Mahjong set contains 144 tiles divided into three categories: suited tiles, honor tiles, and bonus tiles.

Suited Tiles — 108 tiles

Three suits of nine ranks each, with four identical copies of every tile (4 × 27 = 108).

Honor Tiles — 28 tiles

Four winds and three dragons, four copies each.

Bonus Tiles — 8 tiles

Four Flowers (Plum, Orchid, Chrysanthemum, Bamboo) and four Seasons (Spring, Summer, Autumn, Winter). These tiles are set aside when drawn — they score bonus points and the player draws a replacement tile immediately. They are never played as part of the standard hand.

Play proceeds counter-clockwise starting with East. On each turn a player draws a tile from the wall, then discards one tile face-up to the center. The discard is announced aloud. Any opponent may then claim the discard under specific conditions, interrupting normal play.

Claiming a Discard

Three types of claims exist, listed in descending priority:

Chow (食, Sik)

Claim a discard to complete a sequence (three consecutive tiles of the same suit). Only the next player in turn order may Chow. The claimed tile and the two matching tiles from your hand are placed face-up as a meld.

Pung (碰, Pung)

Claim a discard to complete a triplet (three identical tiles). Any player may Pung, overriding a Chow from the next player. The claimed tile and two matching tiles from hand are placed face-up.

Kong (槓, Gong)

Claim a discard — or use a self-drawn tile — to form a quad (four identical tiles). There are three types:

• Exposed Kong: call a discard when you hold three matching tiles in hand.
• Added Kong: add a self-drawn tile to an existing exposed Pung.
• Concealed Kong: form a quad entirely from self-drawn tiles; tiles are placed face-down.

After any Kong, the player draws a replacement tile from the dead wall and must discard before play continues. A Kong adds one tile to your meld without changing the count of tiles in hand.

Mahjong (胡, Wu)

Declare victory. You may win by claiming a discard from any opponent, or by drawing the winning tile yourself (self-draw, 自摸 zimo). Self-draw wins typically score higher.

Wall Exhaustion

If all drawable tiles are consumed without a winner, the hand is a draw (荒牌 fong pai). East redeals and no points change hands. The dealership does not pass.

The Standard Hand

A winning hand contains exactly 14 tiles arranged as four melds plus one pair. A meld is either:

• Sequence (順子 shunzi): three consecutive tiles of the same suit — e.g., 4–5–6 Bamboo.
• Triplet (刻子 kezi): three identical tiles — e.g., three West Wind tiles.
• Quad (槓子 gangzi): four identical tiles, formed via a Kong declaration.

The pair (雀 jeuk) — two identical tiles — functions as the hand's "eye" and cannot be a meld.

Special Hands (特別牌型)

Beyond the standard four-meld-plus-pair structure, Hong Kong rules recognize several limit hands — rare arrangements that score the maximum regardless of their component faan. These are worth a set number of points (typically 64× or 3× the base stake), declared and verified before payment is made.

Thirteen Orphans (十三幺)

One each of: 1 Bamboo, 9 Bamboo, 1 Character, 9 Character, 1 Circle, 9 Circle, East, South, West, North, Red Dragon, Green Dragon, White Dragon — plus one duplicate of any of those thirteen tiles. A fully concealed hand; cannot win on a discard in many rule sets.

Seven Pairs (七對子)

Seven distinct pairs. Must be seven different pairs — no four-of-a-kind counts as two pairs. Concealed; typically worth 3 faan plus any matching bonuses.

Nine Gates (九蓮寶燈)

1-1-1-2-3-4-5-6-7-8-9-9-9 in a single suit, plus any one tile of that suit. Fully concealed. Among the most celebrated hands in all of mahjong.

Four Kongs (槓上槓)

All four melds are quads. Extraordinarily rare due to the draw requirements — each kong triggers a dead-wall draw, making hand assembly a race against wall depletion.

Heavenly Hand (天糊)

East wins with the initial 14 tiles before anyone discards. A once-per-game occurrence that ends the hand immediately.

Earthly Hand (地糊)

Any non-East player wins on East's very first discard, before drawing any tile themselves.

Hong Kong Mahjong uses a faan (番) system. The winner counts the faan their hand earns; the base payment is then doubled for each faan, up to a house-agreed maximum (typically 3 or 5 faan).

Common Faan Values

Pattern	Faan	Notes
Self-draw win (自摸)	1	All three losers pay the winner
Concealed hand on discard (門前清)	1	No exposed melds before the winning discard
All-Pung hand (對對糊)	3	All four melds are triplets or quads
Mixed suit (混一色)	3	One suit + honors only
Pure suit (清一色)	7	Single suit, no honors
Wind pung — seat or round wind	1	Triplet of your seat wind or the current round wind
Dragon pung	1	Per triplet of 中, 發, or 白; stackable if you hold two or three
Three concealed pungs	2	Three hidden triplets in a single hand
All terminals & honors (混老頭)	4	Only 1s, 9s, winds, and dragons
Seven Pairs (七對子)	3	Flat 3 faan; always concealed
Last tile win (海底撈魚)	1	Win on the very last tile in the wall
Win on a kong draw (槓上花)	1	Win on the replacement tile after a kong
Robbing a kong (搶槓)	1	Win by claiming the tile added to an exposed pung
Flower — own seat	1	Flower or season matching the player's seat number
All four flowers / all four seasons	Limit	The complete set of one bonus category

Payment Rules

When the winner claims a discard, the discarder pays double the base payment while the other two pay the single base. When the winner self-draws, all three losers pay double. East always pays and receives double — as dealer, stakes are always higher.

Faan cap and limit hands

Most house rules cap faan at 3 or 5. Limit hands bypass the cap and pay a fixed maximum — agree the limit value before play begins. Common limits: 64 units, or "triple the cap payment."

Doubling Order

Count all applicable faan, then apply them as multipliers to the base unit:

• 0 faan — minimum hand, base payment only (some groups require at least 1 faan to win)
• 1 faan — base × 2
• 2 faan — base × 4
• 3 faan — base × 8
• Cap (e.g. 5 faan) — base × 32 or agreed limit

Additional Rules

Discard restriction after Chow

After claiming a Chow, you may not immediately discard the tile that the next player would need to win (食雞打雀). This prevents a mechanical cycle where one player's Chow feeds another player's winning tile.

Declaring false mahjong

If a player incorrectly declares mahjong with an invalid hand, they pay a penalty to all three opponents equal to the agreed limit hand payment, and the hand continues without that player's participation until the next deal.

Dead hand

A hand is voided (死糊) if the player exposes tiles they should not, melds incorrectly, or acts out of turn. The hand continues; the voided player draws and discards normally but cannot win that hand.

East's privilege

When East wins, the deal does not pass — East deals again with an additional bonus applied to the next hand. East's winning streak earns increasingly large bonuses in many rule sets (the 連莊 system). East passes only when a non-East player wins.

Faan scoring doubles your payment for each faan (a scoring double), so your decisions turn on two things. First, you must clear the table minimum at all, because a 0-faan chicken hand cannot win on a discard. Second, you want to cross a real multiplier threshold before the house cap stops the value from growing. Most hands are won by the player who knew their guaranteed faan from the first tile and did not build past what they could collect.

What to build — find your faan floor first

Before chasing a shape, check what scores even if the hand barely comes together. A dragon pung (a triplet of dragons) or a seat-wind or round-wind pung gives a 1-faan floor on its own and wins on a discard. A clean two-suit sequence hand can finish at 0 faan and only wins by self-draw (+1) or by staying concealed (門前清 +1). Pick a floor, then build on top of it.

• Honour pungs add faan with no conditions. Dragon and seat-wind or round-wind pungs are +1 faan each and put no constraint on the hand shape. A single dragon pung clears a 1-faan minimum and turns a chicken hand into one that pays. A wind that is both your seat and the round counts twice (+2).
• A pung-heavy draw is better built as all-pung than as sequences. 對對糊 is a 3-faan floor you can reach from a pung-heavy draw without committing to a suit, and it adds cleanly: with three concealed pungs (+2) it reaches 5, and with a mixed flush (3+3) it reaches 6.
• Commit to a suit early only if you are aiming for a flush. The jump from 3 to 7 (mixed flush to full flush) is four extra doubles, which multiplies the same 14 tiles by 16. Keep your honours and one main suit alive through the early draws, then decide. Settle for mixed flush (3) if your honours pair into pungs. Drop the honours for pure flush (7) only when your suit count grows past about nine tiles and your wait stays wide.
• Stay concealed when you are one above the floor. 門前清 (+1 on a discard) plus self-draw (+1) clears a 1-faan table with no flush or pung structure at all. Break concealment only to claim a scoring pung you would otherwise miss. Do not break it to speed up a 0-faan shape, because that gives away your last route to the minimum.

Speed vs value — and where the cap kills value

Chase value only when the climb crosses a real multiplier threshold and lands under the cap. Going from 0 to 1 just unlocks the win. Going from 混一色 (3) to 清一色 (7) is worth accepting a shorter wait. Once you can already reach the cap, extra faan adds nothing, so switch to the fastest wait, meaning the hand that is one tile from a win. On a 3-cap table a single 3-faan source (all-pung or mixed flush) already maxes you out, so there is no reason to combine all-pung with a flush; just win first. When the wall thins to a turn or two, drop to the widest 1-faan wait no matter what. A 1-faan win banks real doubles, while a 7-faan wait that never completes scores zero and risks dealing in.

Seat order changes this. The last seat to draw each go-around has fewer live draws ahead, so commit to a value plan only when you are already one tile short with a wide wait. Earlier seats can afford one more turn of drawing for value. As dealer, lower your bar hard. Any win, even a bare 1-faan self-draw or 門前清, keeps the deal and forces everyone to pay you double again next hand. Because East pays and receives double on every hand, that repeat usually outweighs the one or two doubles that slow building would add, so take the fast cheap win the moment you are legal.

Defence — price the threat, then read the discards

On defence, read exposed melds for value before you fold. Two same-suit chows plus an honour pung means the player is going for a mixed or full flush (3/7), so stop feeding that suit and release only off-suit tiles. Scattered suits with no honours usually cap at a cheap 1-faan win, so keep pushing; dealing in costs little and folding gives up a live hand. Denial works best against the dealer, who collects double and repeats. Winning a quick 1-faan off any seat to end the hand before a 7-faan hand lands is worth the small loss.

• An exposed suit means every tile of that suit is dangerous. Against a suspected full flush, throw no tile of their suit even if it looks safe by position, because flush hands win on wide, hard-to-read waits. Honours that cannot extend the flush are the safe tiles to release.
• Read which end of the number tiles they are shedding. A player cutting only middle tiles (4-5-6) is going for all-terminals-and-honours (4 faan), so the safe set flips: middles become safe, while 1s, 9s, winds, and dragons become dangerous. Do not throw 'useless' honours late by reflex, because that is exactly what completes their hand.
• Save genbutsu for the fold. Any tile already in an opponent's own discards is safe against that player; an honour seen three times with no pung claimed is dead. Hold a couple in reserve. Against a concealed hand chasing 門前清 or self-draw, known-safe tiles are your only real read.
• Watch the silent player. No claimed melds by mid-hand means the player is either hopeless or stacked for concealment (three concealed pungs +2, seven pairs 3, or a concealed all-pung at 3). These hands self-draw often, and a self-draw makes all three players pay double. Folding cannot fully protect you, so switch to defence early.

Where the value is

Source	Faan	Why it matters for decisions
門前清 + self-draw	1 + 1	'Free' faan at zero tile cost; self-draw also makes all three pay double. Protect them — don't call cheap melds.
Dragon / wind pung	1 each	Adds faan with no shape constraint. Cheapest path off a chicken hand; a seat-and-round wind counts +2.
All-pung (對對糊)	3	Cheapest 'real' floor; reaches 6 with a mixed flush and 5 with three concealed pungs. Pivot here on a pung-heavy draw.
Mixed flush (混一色)	3	Strong and far easier than full flush. The default flush target on most tables; already maxes a 3-cap table.
Full flush (清一色)	7	The jump from 3 to 7 multiplies by 16. Worth a shorter wait, but it overshoots a 3-cap table, so chase it only where the cap is 5+ and the wall is full.
All terminals & honours	4	Near-cap; reachable when your pungs are all 1s/9s/honours. Note its flipped defensive tells (middles safe, ends deadly).

Hong Kong mahjong combines two math problems. The first is a hypergeometric drawing process: how likely is the tile you need. The second is a heavy-tailed payoff function: how much the hand is worth if that tile lands. Almost every decision comes down to one of these two questions. A little combinatorics helps with both.

The drawing model

You can see your own 13 tiles and every face-up tile. Call everything else unknown. Suppose u tiles are unknown and your hand is waiting on w of them. Those w tiles are the ones that would complete your hand. Each unknown tile you turn over is a winner with probability w/u. A drawn tile does not come back, so you sample without replacement.

What are the odds of hitting at least one of those w winners over your next d draws? The clean way is to count the ways you miss them all. Write C(n, k) for the binomial coefficient, read “n choose k”. It is the number of distinct groups of k tiles you can pick from n when order does not matter. For example, C(5, 2) = 10: there are ten ways to choose 2 things from 5. Two of these counts matter here:

• C(u, d) — the number of equally-likely ways your d draws could come out of the u unknown tiles.
• C(u−w, d) — how many of those draws avoid every winner, coming entirely from the u−w losing tiles.

Their ratio, C(u−w, d) / C(u, d), is the chance you draw a blank all d times. The chance of completing is one minus that: 1 − C(u−w, d) / C(u, d). This is the hypergeometric distribution, the exact law for drawing without replacement. When the winners are only a sliver of what is left, written w ≪ u and read “w is much smaller than u”, pulling a few tiles out barely changes the odds. Each draw then misses about independently with probability 1 − w/u, and the chance of d misses in a row is that figure multiplied by itself d times: about (1 − w/u)^d. That gives the easy approximation 1 − (1 − w/u)^d, the same expression you would use for coin-flips or for drawing with replacement.

Two Hong Kong facts change the result. You may declare a win on any player’s discard, so a single go-around can expose you to as many as four unknown tiles rather than one. That is a ceiling, though, since a wary opponent who reads your wait will not discard the tile. The wall also yields only about 18–19 draws per player: 91 tiles remain after the deal, minus the kong/flower reserve. So d is small. You do not get many tries, which is why efficiency matters.

A worked example

Suppose your hand is one tile from winning (it is ready) on a two-sided wait, so w = 8 tiles complete it and none are yet showing. Say u = 100 tiles are unknown. Each draw is a winner with probability 8/100 = 8%. The chance of drawing at least one winner over your next d draws is 1 − C(92, d) / C(100, d):

• after d = 6 draws: about 40%;
• after d = 12 draws: about 65%;
• after d = 18 draws, roughly a full hand: about 81%.

The chance builds in an S-shape: about 8% after one draw, 50% by the eighth draw, and 81% by the eighteenth. These numbers count only your own draws. Because you may also win on any opponent’s discard, the true chance is higher, by an amount that depends on how freely opponents release your tile.

Your average winning draw

For a wait on w tiles among u unknown ones, the expected position of the first winner has a simple closed form: (u + 1) / (w + 1). With u = 100:

• a two-sided wait (w = 8) lands on about the 11th draw, well inside a hand’s ~18 draws;
• a closed or edge wait (w = 4) lands on about the 20th draw, past the end of most hands, so these waits often need a discard to complete;
• the Nine Gates (w = 23) lands on about the 4th draw.

Halving the number of winning tiles roughly doubles how long you wait.

Drawing without replacement helps you

The exact figure of 81% over 18 draws is a little higher than the coin-flip estimate 1 − (1 − 0.08)^18 ≈ 78%. The reason is that tiles do not come back. After a draw that misses, one losing tile is gone, so the next draw wins with slightly better odds: 8/100 = 8.00% on the first draw, and 8/99 ≈ 8.08% on the second if the first missed. Sampling without replacement works a little in your favour, and the gap from the coin-flip estimate grows with the number of draws.

Wait width is the lever you control

The number w depends entirely on the shape of your wait, and the spread is large:

Wait	Example	Winning tiles w
Two-sided (兩面)	4-5 → 3 / 6	8
Edge (邊張)	1-2 → 3	4
Closed (嵌張)	4-6 → 5	4
Dual-pair (雙碰)	two pairs → either	4
Single (單吊)	pair your last tile	3
Pure Nine Gates (九蓮)	1112345678999 → any 1–9	23

That last row is the widest legal wait in the game. The hand 1112345678999 completes on all nine ranks, leaving (4−3) + 7·(4−1) + (4−3) = 23 winning tiles. As a concrete example, with u ≈ 100 a fully-live two-sided wait (w = 8) is 8% per draw and 81% to self-draw across a hand’s ~18 draws. Both figures are exact, since your own draws are a uniform sample. Counting opponents’ discards as well raises a single go-around toward 28%, but treat that as a blind-discard upper bound. A closed wait (w = 4) is almost exactly half: 55%. Half the winning tiles gives about half the chance, and that is why tile efficiency matters.

The opening hand

The dealt 13 tiles, drawn from the 136 playing tiles (flowers aside), give a few exact probabilities:

• You almost always start with a pair. The chance all 13 tiles are different kinds is C(34,13)·4^13 / C(136,13) ≈ 12.9%, so you hold at least one natural pair about 87% of the time.
• Flowers are a coin-flip. P(no flower) = C(136,13) / C(144,13) ≈ 46%, and you expect 13·8/144 ≈ 0.72 of them.
• The space is large but not absurd. There are C(136,13) ≈ 4.8 × 10^17 distinct opening hands. That is enough that you will never repeat one, and small enough that the events above have everyday probabilities.

Why the game has a cap

A hand is worth base × 2^faan. Faan is the unit of hand value. The payoff doubles with each faan, while each additional faan is progressively rarer. This is the same multiplicative shape as the St. Petersburg game. Whether the raw expected payoff truly diverges depends on how fast high-faan hands thin out. The house cap (usually 3 or 5 faan) removes the question: it bounds the payoff (at most base × 2^cap), making the hand’s value a bounded random variable. Its mean, variance, and every moment beyond them are finite, so the game stays playable.

The cap also changes the objective at a sharp boundary. Below the cap, every extra faan doubles your money. At or above it, the next faan is worth exactly zero. So the rational player maximizes value until the cap is in hand, then switches to maximizing the probability of finishing at all: stop building, and just win.

Discards are evidence

Every discard is a sample that updates your estimate of an opponent’s hand. The standard safe-tile heuristic suji (筋) is one example: if a player discarded a 4, they were not holding the two-sided wait that needed it, so the 1 and 7 on that suit-line are measurably safer to release. Reading the table is Bayesian updating, with tiles as the data.

Term	Cantonese	Meaning
Mahjong	麻雀 Maa Jeuk	The game; also the winning call
Faan	番	Scoring double; the unit of hand value
Pung	碰 Pung	Triplet; to claim a discard for a triplet
Chow	食 Sik	Sequence; to claim a discard for a sequence
Kong	槓 Gong	Quad; to form or claim a set of four
Zimo	自摸	Self-draw win
Wu / Hu	胡	Win; mahjong declaration
Jeuk (pair)	雀	The pair / "eye" of the hand
Fong pai	荒牌	Draw — wall exhausted, no winner
Dead wall	嶺上	Reserve tiles for bonus/kong draws