Quick answer
Each dragon belongs to a suit. Red goes with craks, green with bams, and the white dragon, the soap, goes with dots. They are never interchangeable, so a soap cannot be played as a red or green dragon. A soap counts as a zero only where the card shows one, and then it belongs to no suit. A joker can fill a pung, kong or quint of dragons, but never a single dragon or a pair.
Dragons look decorative and are not, which is where the confusion starts. Each one belongs to a suit, and once you know which goes with which, a whole set of card lines stops looking arbitrary. The rule is short. Red goes with craks, green with bams, and white with dots.
The twelve tiles
There are three kinds of dragon and four of each, so twelve in a set. Red, green, and white. The white dragon is also called the soap, named for a face that is blank or lightly framed and looks like a bar of soap.
On the card a dragon is written as a D, and its colour works like every other colour on the card. The colours are relative, not fixed, so a green D does not mean a green dragon. A D printed in the same colour as a group of numbers is the dragon of whatever suit you play that group in. A D printed in a different colour is the dragon of a different suit.
Each dragon belongs to a suit
This is the rule that makes those card lines readable, and it is easy to play for months without noticing it. Red dragons go with craks. Green dragons go with bams. White dragons, the soaps, go with dots.
So when a card line prints its dragons in the same colour as its numbers, the dragons are not free choice. They are the dragons of that suit. A hand of bams wants green dragons. A hand of craks wants red. Get this wrong and you build a hand that looks finished and is not.
Matching and opposite dragons
You will hear both phrases at the table, and the colours on the card settle them. Matching dragons are printed in the same colour as the numbers they go with, so they must be the dragons of the suit you play those numbers in. Bams take greens, craks take reds and dots take soaps.
Opposite dragons, also called opposing dragons, are printed in a different colour from the numbers, so they must be the dragons of a different suit. With bam numbers, opposite dragons are reds or soaps, never greens.
Dragons are not interchangeable
To be plain about it, a soap may not be played as a green or a red dragon, and it may not stand in as a dot tile inside a run of numbers. Each dragon is its own tile, and the suit it belongs to does not make it a member of that suit.
In practice, three dragons of mixed colours are three unrelated tiles, not most of a pung. Holding one of each does not bring you any closer to a dragon pung.
When a soap becomes a zero
The soap has a second job. On card lines that need a zero, the year hands most obviously, the white dragon stands in for it, because a mahjong set has no zero tile and years since 2000 keep needing one.
Two conditions matter. First, a soap is only a zero when the card shows it that way, not whenever you would find one convenient. Second, when it is acting as a zero it is neutral. It stops being the dot dragon and belongs to no suit at all. That is why a hand can pair a soap-as-zero with numbers from any suit without breaking the suit rule.
One limit follows from this. A soap acting as zero cannot be part of a run of consecutive numbers, because runs are built from the number tiles one through nine and a zero is not among them.
Dragons and jokers
Dragons follow the ordinary joker rule rather than a special one. A joker can stand in inside a pung, kong or quint of dragons, and can never stand in for a single dragon or inside a pair of them.
That matters because card lines can ask for a pair of dragons rather than a pung. A pair of soaps is a pair like any other. No joker can fill it, and a discarded soap can be called for it only when that soap is the last tile your winning hand needs. Otherwise both soaps have to reach you in the deal, the passes or your own draws.
Playing them
Early, a single dragon is a weak holding. It serves only the lines that want its exact colour, and until you have chosen a direction it is a tile with one job and no evidence that job exists. Most of the time it is a comfortable pass.
Two of the same dragon are different, because a joker can never fill a pair, and a natural pair of dragons opens the winds and dragons lines and the suit lines that call for that dragon. Hold them.
Late, a dragon can be a dangerous discard. An opponent showing an exposure of numbers in one suit may want that suit's dragon or, on a line with opposite dragons, one of the other two. So if somebody has bams exposed, a green is one candidate for them, and a red or a soap is not ruled out.
What the dragon symbols mean
The Red Dragon carries the Chinese character 中, which means middle or center. The Green Dragon carries 發, short for a phrase that means to strike it rich. Some sets draw a red or a green dragon in place of the character.
The White Dragon is the soap. Its Chinese name means white board, and the tile is either blank or framed with a border. Some sets print a P in the middle of it, from an older spelling of the Chinese word for white.
Corner letters come from the same words. A C on the Red Dragon is from chung, an older spelling of 中, and an F on the Green Dragon is from fa. That F belongs to the dragon. On the card, an F stands for a flower.
None of this changes how a dragon plays. At the table the colour is what counts, because the colour decides the suit. The red goes with craks, the green with bams and the soap with dots, whatever is printed on the face.
Put it into practice
New to the game? Start the course at Lesson 1, Meet the Tiles. The first three lessons are free and need no account.
Already play? Have the coach read a real dealt hand for you, free. It names the direction the hand goes for, what to keep, what to pass, and the reasoning behind each call.
Knowing which dragon belongs to which suit is the easy half. The harder half is spotting, in your own thirteen tiles, which lines that unlocks. Mei, our AI coach, reads a dealt hand and says which direction it goes for and what to keep, dragons included.