Chinese Classical Mahjong

Chinese Classical mahjong is the ancestral form, sometimes called the “old Chinese” style. It is built like any mahjong. A winning hand is four melds and one pair, made from the standard set of 144 tiles. A meld is a set of three or four matching tiles, or a run of three. What sets the classical game apart is how a finished hand is scored and paid. This is the older, fuller scoring system. Joseph Babcock introduced it to the West in the early 1920s with his Rules of Mah-Jongg (the famous “red book”), and A. D. Millington later wrote it up in The Complete Book of Mah-Jongg. The British Mah-Jong rules follow this lineage. The faster scoring of modern Hong Kong play is a streamlined descendant of it.

The defining trait is points plus doubles, capped by a limit. You first add up base points for the win itself and for each scoring element — pungs (three of a kind), kongs (four of a kind), valuable pairs, flowers and seasons. Then certain patterns each double the running total: dragon and wind triplets, a one-suit hand, a hand with no sequences, and so on. The doubled figure is then capped at an agreed limit (classically 500 points; many tables use 1,000). A few rare hands score the full limit outright.

Settlement is richer too. All three opponents pay the winner, and the dealer (East) both pays and receives double. In full classical play the three losers also settle the differences between their own hands, scoring their own (unfinished) hand — adding up whatever scoring sets they did manage to assemble — so even a hand that did not win can earn or owe. Hong Kong keeps the points-and-doubles skeleton but trims most of this away. That is why the two games feel related but are not identical.

How it differs from Hong Kong

A Chinese Classical 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). Within a suit, the 1 and 9 are terminals (the end ranks) and the 2 through 8 are simples.

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.

The deal uses the familiar mahjong setup — all 144 tiles built into a square wall, dealt to four players, each building toward a hand of four sets plus a pair. Two parts of the setup matter more here than in faster play: the prevailing (round) wind, which feeds directly into scoring, and the loose tiles set aside to replace kongs and flowers.

The four-round game and the prevailing wind

A full game runs four rounds, each named for a prevailing wind in the fixed order East → South → West → North. Within each round every player serves at least once as dealer, so a round is at least four hands and a complete game is 16 hands or more — it stretches whenever the dealer wins or a hand washes out (a wash-out is a hand that ends with no winner because the tiles run out).

The prevailing wind affects scoring. A pung or kong of the round wind is worth one double, exactly like a pung of your own seat wind. In the Millington/British school, if your seat wind and the round wind are the same tile (you sit East in the East round), that single triplet can earn two doubles — one for seat, one for round.

Dealer and how the deal passes

1. Seat the winds. Dice (with a draw of the four wind tiles) settle who starts as East. East is the dealer; the others take South, West, and North going counter-clockwise.
2. East keeps the deal on a win — and, in the classical/British school, on a wash-out. If the dealer goes Mah-Jong, East stays East and an extra hand is played. Most classical tables also keep East on a drawn ("goulash") hand with no winner; seats do not move.
3. Otherwise the deal passes counter-clockwise. When any other player wins, the deal moves on: the player who was South becomes the new East. When the deal has cycled all the way around the table, the prevailing wind advances to the next round.

The dead wall and loose tiles

After breaking the wall, a block of tiles at the broken end is fenced off as the dead wall (the "kong box"). These are the loose tiles — the supply you draw a replacement from every time you declare a kong or expose a flower. You never draw your normal turn tile from here, and the +2 self-draw bonus (the reward for drawing your own winning tile from the wall rather than claiming a discard) is only for the winning tile taken from the live wall, not from this box.

How flowers and seasons score

In the classical game the eight flowers and seasons score points, so they are always declared face-up.

• Set out at once. Any flower or season in your dealt hand is laid face-up beside your hand immediately, and you draw a loose tile to replace it. The same happens the moment you draw one during play.
• Every bonus tile scores 4 points just for being shown.
• Match your seat number for a double. Flowers and seasons are numbered East = 1, South = 2, West = 3, North = 4. In the British and most common scoring, the flower (or season) bearing your seat's number earns one double each — and because your seat changes as the deal moves, a flower that scores a double this hand may score none next hand.
• Collecting a full set — all four flowers, or all four seasons — is worth a further double in Millington's own scoring (the British school scores a complete set as two doubles).

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. Each one completes a meld — a finished group of tiles (a sequence, triplet, or quad) laid face-up for all to see:

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 (a completed set revealed on the table).

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 (one you draw yourself rather than claim from a discard) — to form a quad (four identical tiles). There are three types:

• Exposed Kong: call a discard when you hold three matching tiles in hand. (It is exposed — built from a claimed tile and shown face-up.)
• 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. (It is concealed — built only from tiles you drew, never claimed.)

After any Kong, the player draws a replacement tile from the dead wall (the reserved tiles at the wall's end, set aside as a replacement supply) 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.

A hand that is one tile short of completing is ready (Japanese tenpai); the tiles that would finish it are your wait. A wide wait has many such tiles; a narrow one, few.

Limit hands

Separately, a handful of rare patterns are so prized that they pay the full limit outright, regardless of how few points they would otherwise count. These are the limit hands. Unless noted, each is still a normal four-sets-plus-a-pair hand; several are required to be concealed.

A note on the two trickiest. Thirteen Orphans is one of every terminal and honour — the 1 and 9 of each suit, each of the four winds, and each of the three dragons — with one of those tiles duplicated to make the pair. Nine Gates is a single suit holding three 1s, three 9s, and one each of 2 through 8, won on any further tile of that suit.

A few related patterns sit just below the top tier. Little Three Dragons (two dragon pungs + a pair of the third) is a strong doubling hand but not a limit hand. And some British-style rule sets score certain hands as a half-limit (250 of a 500 limit) rather than the full amount — Earthly Hand is the standard example in Millington, and some tables also half-score a plain one-suit flush (Purity / 清一色) or other near-misses.

Base points

In Chinese Classical scoring you count base points first, then multiply by doubles. The base points are what the winner adds up before any doubling. Three rules explain almost every number below. A concealed (built in your hand, kept hidden) set outscores an exposed (claimed from a discard and laid face-up) one. Terminals and honours (1s, 9s, winds and dragons) outscore the simple 2–8 tiles. And a kong scores roughly four times its matching pung.

Points for sets

A kong claimed by adding a fourth tile to an exposed pung counts as exposed; only a kong assembled wholly from the wall and your own hand counts as concealed. The exposed/concealed distinction is also why you keep a concealed kong face-down until the hand ends. You may declare it only to draw the replacement tile, and it still scores at the concealed rate.

Pairs, flowers & the win itself

• Pair of dragons — 2 points.
• Pair of your seat wind — 2 points; pair of the round (prevailing) wind — 2 points. A pair that is both your seat wind and the round wind scores 2 + 2 = 4 at most tables (some score only 2).
• Each flower or season tile — 4 points apiece. They are set aside face-up and replaced with a fresh draw the moment they appear, then scored at the end.
• Going Mah-Jong (completing the hand) — 20 points to the winner only (Millington / British / North American value; some older tables use 10).
• Ordinary pairs of simples, and a pair of a wind that is neither your seat wind nor the round wind, score nothing.

Winning bonuses

These extra base points go to the winner only, added before doubling. They are not mutually exclusive — a self-draw that is also the only fitting tile earns both +2s.

• Self-draw (you draw your own winning tile from the wall rather than claiming a discard) — +2.
• Only-tile wait — the tiles your hand still needs to finish are called your wait, and this is winning on the single tile that can complete the hand: an edge wait (1–2 waiting 3, or 8–9 waiting 7), a closed/middle wait that waits on a single middle tile, like 4–6 needing the 5, or a single-tile pair wait — +2.
• Out on a loose / supplement tile (the replacement tile drawn from the dead wall after a kong) drawn to replace a tile after declaring your own kong — +2.
• Robbing a kong — winning on the tile an opponent adds to promote an exposed pung to a kong — +2.

Doubles

Base points get you a running total. Doubles are the multiplier layer that sits on top of it. Where base points reward the building blocks of a hand — pungs, kongs, the pair, flowers — doubles reward shape: a one-suit hand, a hand with no sequences, or a pung of your seat wind.

Each double does exactly what the name says: it doubles your entire running total. They stack, so a hand worth two doubles is multiplied by four, three doubles by eight, four doubles by sixteen, and so on. A few doubles can take a 30-point hand to the limit.

Order of operations

Sum every base point first, then apply the doubles to that total — once each, all at the end. Do not double individual scoring items as you go.

1. Add up all base points: 20 for going Mah-Jong, plus pungs, kongs, the pair, flowers, and any winning bonuses.
2. Count how many doubles the hand earns from the table above.
3. Apply them to the total: total = points × 2^{(number of doubles)}.

So two doubles multiplies by 4, three by 8, four by 16. The doubles are applied to the whole base total at once — not item by item — which is exactly why one or two of them swing a hand so hard.

The seat-and-round wind variant

One rule worth pinning down before you sit down: a single wind can be both your seat wind and the prevailing round wind — for example, you are sitting East in the East round. Most classical tables, including the British Mah-Jong rules, let a pung of that wind score 2 doubles: one for the seat, one for the round. A minority of rule sets count it only once. Agree which before play, because it routinely decides whether a hand reaches the limit.

The limit

In Chinese Classical scoring, a single hand is worth its base points multiplied by two for every double it earns — the running total is points × 2^doubles. These totals grow fast: a full flush with several dragon pungs can run into the thousands. To keep settlement sane, the table agrees in advance on a limit — the most any one hand can ever pay out. Any hand whose score meets or exceeds the limit simply scores the limit, and no further.

Classical settlement differs from faster games in three ways: everyone pays the winner, the dealer's payments and receipts are both doubled, and in the fullest version the losers also settle with each other. This three-part settlement is the main thing modern Hong Kong play dropped.

The core rule: all three losers pay the winner

When a player goes Mah-Jong, each of the other three players pays the winner the full value of the winning hand — not a split, not a share. If the hand is worth 40 points, the winner collects 40 from each loser, for 120 in total. In casual play, often only the discarder pays. (The winner always takes the round, even if a loser's own hand happened to score more.)

The dealer (East) doubles — both directions

Every payment to or from East is doubled:

• If a non-dealer wins: the two non-dealer losers each pay the hand's full value, but East pays double that value.
• If the dealer (East) wins: every loser pays double the hand's value, so East's winning hands pay out twice as much.
• The doubling also applies to the loser-to-loser settlement below: any difference paid to or from East is doubled.

The fuller method: losers settle with each other

Traditional Classical settlement does not stop at paying the winner. Each of the three losers also scores their own (unfinished) hand — counting their pungs, kongs, flowers, doubles, and so on, but omitting the 20 for going out and the win-only bonuses, since they did not win. Then every pair of losers exchanges the difference of their two totals: the lower hand pays the higher hand the gap between them, with East's share doubled as always.

This rewards building a strong hand even when you do not win, and it is the historically correct method (Millington's rules call it out as the full Classical procedure). Many modern tables drop it and pay only the winner — both are in common use, so agree before you start.

Discarder Responsibility — Pao (包)

For the largest limit hands, fuller classical (and Chinese Official) rules add a penalty for feeding the killing tile. Under Pao (包, "liability"), if an opponent has already exposed two dragon pungs/kongs and you discard the third dragon to complete Big Three Dragons (大三元) — or has three wind pungs exposed and you feed the fourth wind for Big Four Winds (大四喜) — you alone pay the entire limit for all three losers. The other two pay nothing. The reason: with the giant hand already exposed, the player who supplies the final tile takes responsibility for it.

The trigger is that the big hand must have been visible (its other dragons or winds already melded face-up) at the moment you discarded; a Pao hand completed by self-draw, or one assembled concealed, is settled normally. Plenty of casual Classical tables omit Pao entirely.

And all of it is bounded: the limit caps the value of any single hand (classically 500 points; many tables use 1,000, and some English clubs 2,000) before payment. The cap is applied to the hand first, then the East doubling is layered on top — so a dealer can still collect up to twice the limit from each loser.

Most of your score comes from doubles, not base points. A double multiplies your whole total. A point adds a little. Two extra simple pungs add four points. One flush multiplies everything by four. Before any move, ask whether it earns another double or only adds points. Build toward doubles, stay concealed long enough to keep the concealed-hand double, and watch the table for the limit hands that can make you pay alone. (A pung is three of a kind; a chow is a run of three in one suit; a double multiplies your whole total by 2.)

What to build: aim for doubles

Commit to one suit early. A full flush is three doubles (×8); a half-flush is one (×2). If by your sixth or seventh draw you hold 7+ tiles of one suit plus honours, discard your weakest off-suit tile (a tile not in the suit you’re collecting) and go for the full flush. Settling for a half-flush to keep a stray chow suit costs you a ×4 jump. Then add more doubles on top of it:

• Pung the right honours, never chow. A chow is 0 points and 0 doubles. A dragon, seat-wind, or round-wind pung is points plus a double each, and using no chows at all earns the all-pungs double too. (A guest-wind pung — a wind that is neither your seat nor the round — pays points but no double, so it helps only through all-pungs or the flush.) Three scoring honour pungs can add 3–4 doubles before the flush even counts.
• Treat your own flower or season as a double, not a bonus. Your own flower is 4 points and a ×2; an off-seat flower is just the 4 points, no double. Once you hold one own-flower double, the cheapest way to the cap is one more double, not another pung for base points.
• When two dragons are paired, take the third over anything. Two dragon pungs plus the third dragon as a pair is already about 3 doubles (two pung doubles plus little three dragons). Completing that pair into the third pung gives you Big Three Dragons, which pays the limit outright.

Speed vs value: stay concealed, and don't claim to hurry

The fully-concealed double is a ×2 on the entire hand. It doubles your base points and every other double you already hold. A flat 30-point open hand is 60 concealed; a hand at ×4 becomes ×8 just by staying closed. That math is why the small tempo loss of passing on a discard is almost always worth it. (A concealed set is one you build in hand; an exposed set is one you claim from a discard, and claiming it ends your fully-concealed double.)

• Don't claim a pung you already hold two of. Claiming the discard halves that triplet's base points (exposed 2 vs concealed 4 for simples; 4 vs 8 for terminals and honours) and locks your hand open, which kills the concealed double on everything. Only claim if it wins the hand outright.
• Hold concealed kongs; don't kong off a discard. A concealed kong is double the base of an exposed one (simples 16 vs 8; terminals and honours 32 vs 16) and keeps your hand concealed. Drawing the fourth tile to declare a concealed kong — and taking the replacement draw — is fine. Exposing a meld to kong someone's discard for the base bump is a bad trade.
• Aim your final wait toward self-draw. Self-draw is +2 points and keeps the hand off the table, which secures the fully-concealed double. Being fed an honour to complete a pung forces you to meld it exposed (base 4 instead of 8) and breaks concealment. So prefer the wait you are more likely to draw than be handed.

Defence: watch for limit hands, and East pays and collects double

One-suit hands often grow to ×8 and beyond and hit the limit. The moment a player is clearly collecting a single suit plus honours, give up the hand: discard off-suit tiles and honours you already hold a pair of, and concede the small loss. The biggest risk is Pao. Feeding the completing tile of Big Three Dragons or Big Four Winds makes you alone pay the whole limit instead of splitting it three ways.

• Read exposed melds before every risky discard. Visible dragon and wind pungs and kongs show you the doubles already locked in. Two dragon pungs down means you never release the third dragon. Hold every copy to the end. The same goes for a third wind once two wind pungs show.
• Block the dealer first. East pays and collects at double the settlement, so the same hand costs you about 2× more. When you can only stop one threat, stop East's. When you must deal in, prefer a non-dealer's modest hand.
• If you can't win, don't finish poorest. Losers also settle the differences between their hands, so a bare 20-point hand pays out to everyone above it. Late in the hand, give up the full-flush plan and grab cheap honour pairs and a flower so your hand isn't the table's lowest.

Where the value is, and when to stop

Doubles compound, so the jumps that matter aren't evenly spaced. The table below shows where the big steps are. Once your points × multiplier already clears the cap, stop optimizing and go out. Extra flowers, self-draw, and the only-tile-wait add nothing past the limit, and over-reaching just risks the hand to an opponent who then pays you zero. Also, don't give up a winning flowered double-stack to chase a named limit hand. The limit is a flat ceiling that flowers and points can't raise, and the named hand is usually far less likely to complete.

If you're East, weight all of this even more heavily. Your deal pays and collects at double the table's settlement, so the dealer seat is the one to over-invest in doubles and never waste on a cheap, fast, double-poor win.

Two layers of mathematics sit under the game. The first is the draw: how likely is it that the tile you need turns up. This works the same in every style of mahjong. The second is the payoff: how much the hand is worth if that tile lands. That is where Chinese Classical differs, because here the score multiplies. A little combinatorics — the math of counting how many ways tiles can be chosen — 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 — statisticians call this drawing 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 \(\binom{n}{k}\) 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, \(\binom{5}{2} = 10\): there are ten ways to choose 2 things from 5. Two of these counts matter here:

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

Their ratio, \(\binom{u-w}{d} / \binom{u}{d}\), is the chance you draw a blank all \(d\) times. The chance of completing is one minus that: \(1 - \binom{u-w}{d} / \binom{u}{d}\). This is the hypergeometric distribution — the exact formula for the odds when you draw without replacement. When the winners are only a sliver of what is left, written \(w \ll u\) 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 (a close estimate, simpler than the exact formula above) \(1 - (1 - w/u)^{d}\), the same expression you would use for coin-flips or for drawing with replacement.

Two facts change the result. You may declare a win on any player’s discard, so a single go-around (one full lap of the table, all four players taking a turn) 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 (you hold two touching tiles like 4-5, so either of two ranks completes it — eight tiles in all), 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 \(1 - \binom{92}{d} / \binom{100}{d}\):

• 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 — slow at first, fastest in the middle, then leveling off: 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 average draw on which your first winning tile shows up has a simple formula you can plug numbers into (a “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 (treating every draw as the same fixed odds, as if the tile went back each time) \(1 - (1 - 0.08)^{18} \approx 78\%\) (the symbol ≈ meaning “is about equal to”). 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 \(8/99 \approx 8.08\%\) 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:

That last row is the widest legal wait in the game. The hand \(1112345678999\) completes on all nine ranks, leaving \((4-3) + 7\cdot (4-1) + (4-3) = 23\) winning tiles. As a concrete example, with \(u \approx 100\) a fully-live two-sided wait (fully live = all eight winning tiles still unseen, so \(w = 8\)) 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 (every unseen tile is equally likely to come up). Counting opponents’ discards as well raises a single go-around toward 28%, but treat that as a blind-discard upper bound (the most it could be, and only if opponents discard carelessly). 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 \(\binom{34}{13}\cdot 4^{13} / \binom{136}{13} \approx 12.9\%\), so you hold at least one natural pair about 87% of the time.
• Flowers are a coin-flip. \(P(\text{no flower}) = \binom{136}{13}\big/\binom{144}{13} \approx 46\%\), and you expect \(13\cdot 8/144 \approx 0.72\) of them.
• The number of possible opening hands is large but not absurd. There are about \(\binom{136}{13} \approx 4.8 \times 10^{17}\) (480 quadrillion) different ones. That is enough that you will never repeat one, and small enough that the events above have everyday probabilities.

Value multiplies, it doesn't just add

Score is \((\text{sum of base points}) \times 2^{\text{doubles}}\). You take the base total and double it once for every “double” the hand earns. Two doubles quadruple it. Three doubles multiply it by eight, and so on. \(2^{\text{doubles}}\) means \(2\) multiplied by itself that many times. A point adds to the total. A double multiplies the total. Two extra simple pungs (a pung is a set of three identical tiles) add \(+4\) points. A single flush adds \(\times 4\) on the entire pile. So a hand’s value depends almost entirely on its number of doubles, not its base points. A full flush (3 doubles, \(\times 8\)) is worth more than any amount of patient pung-stacking you could realistically assemble.

The rare huge hands, the limit, and the swings

Because value scales as \(2^{\text{doubles}}\), the score distribution (the full spread of possible hand values and how often each shows up) is right-skewed (a few very large scores stretch the high end while most hands cluster low) and log-scaled (scores climb by repeated doubling, so equal steps in doubles multiply rather than add). The typical hand — the median, the one right in the middle when you line all hands up by value, is worth less than the average. The average is pulled up by the rare high-double hands. The classical limit (500 points, often 1,000) caps the multiplicative payoff. It cuts off the tail of that distribution (the far high end, where the rare huge scores sit). Once the limit applies, the gap between the largest and smallest hands stays finite and moderate. The big gap between the average hand and the typical (middle) hand is a feature of the scores before the limit is applied. The consequence for play: chasing doubles has positive expected value (it pays off on average over many hands) but high variance (the result swings wildly from hand to hand). So hand selection is a bankroll question (how much you can afford to lose on a bad run) rather than a pure expected-value one: the right call depends on how large a swing your stake can absorb, not on the average payout alone. Go for the big hand when you can afford the swings, and take the sure small one when you cannot.

A simple model shows why the cap matters for risk, not only for size. Suppose each extra double is about a fraction \(q\) as likely as the one before, so the number of doubles falls off geometrically (each step is the same fraction smaller than the last, like 1, then q, then q×q, …). The mean multiplier — written \(E[2^{\text{doubles}}]\), meaning the average value of \(2^{\text{doubles}}\) across all hands — is then finite (it settles to a definite number rather than ballooning without limit) only when doubles thin out faster than half each step (\(q < 1/2\)). The variance depends on the average of the squared multiplier (\(E[4^{\text{doubles}}]\), since squaring \(2^{\text{doubles}}\) gives \(4^{\text{doubles}}\)), which needs the stricter condition \(q < 1/4\). A hand distribution can therefore have a finite average value and still have infinite variance — meaning the swings have no upper limit, because once in a great while a hand is so large it dwarfs everything else, even though the long-run average is an ordinary number. Take \(q = 0.4\): the average hand is worth about ×3 its base points, but the variance is infinite, because rare large hands dominate the spread. Cap the doubles at five and the average barely moves, to about ×2.3, while the standard deviation (a measure of the typical swing around the average) becomes finite, about ×3.8. The limit changes the average payout very little. Its main effect is to make the swings finite, which keeps the game playable.

Discards are evidence

Every discard is a sample that updates your estimate of an opponent’s hand. One standard safe-tile rule, suji (筋), works like this: if a player discarded a 4, they were not holding the two-sided wait that needed it, so the 1 and 7 in that same suit are measurably safer to release. Reading the table means revising your estimate of each hand as the discards come in.

Pao: variance, concentrated

The pao (包) rule applies when you feed the tile that completes a Big Three Dragons or Big Four Winds. You alone then pay the whole limit. Under pao, one player pays the game’s largest payout alone instead of sharing it, which is the worst single outcome a loser can face. In classical defence, never discard the tile that completes an opponent’s third dragon or fourth wind once the rest of that set is exposed.

Tile names

Every tile has an English and a Chinese name. The number tiles are the easy ones: a number tile’s name is simply its number followed by its suit. Learn the nine numbers and the three suits and you can read all 27 of them.

Readings are given as Mandarin / Cantonese — the same characters, two dialects. Mandarin (pinyin: yī, èr, sān…) is the form most learners know; Cantonese (jat, ji, saam…) is what you hear at a Hong Kong table.

The numbers 1–9 are 一 (yī / jat), 二 (èr / ji), 三 (sān / saam), 四 (sì / sei), 五 (wǔ / ng), 六 (liù / luk), 七 (qī / cat), 八 (bā / baat), 九 (jiǔ / gau). The three suits are:

Put any number in front of any suit and you have the tile:

Honour tiles: winds & dragons

The honour tiles carry no numbers. Each one’s name is the character printed on it.

Bonus tiles: flowers & seasons

Eight bonus tiles sit outside the three suits. They never form part of a hand; when you draw one you set it aside for a small score and draw a replacement.

Game terms