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Mahjong, like gin rummy, is about drawing a tile and discarding a tile while collecting sets.
All you need in Mahjong is to assemble 4 sets of 3 tiles and one pair (called eyes).

Sets can be a sequence of 3 (chow) or 3 of a kind (pung) like in a picture below.

pung
  3 identical tiles
example: Mahjong Bamboo #4 Mahjong Bamboo #4 Mahjong Bamboo #4
chow
  3 same-suit consecutive tiles
example: Mahjong Character #1 Mahjong Character #2 Mahjong Character #3
   pair
  2 identical tiles
example: Mahjong Circle #3 Mahjong Circle #3
There's also a name for an extended pung - a kong
A kong is a set with 4 identical tiles, e.g.:  
Mahjong Bamboo #4 Mahjong Bamboo #4 Mahjong Bamboo #4 Mahjong Bamboo #4

Under Hong Kong Mahjong rules, players can claim a just discarded tile for a chow only from the player to their left.
If the tile being discarded is not claimed right away, it becomes "dead" and it can't be used later to build sets.

Players can claim a just discarded tile for a pung (3 of a kind) from any player, not necessarily from the player to their left.

As with a pung, players can claim a just discarded tile needed for a win. It can be claimed from any player - either for a chow, or for the eyes, or for a pung.

You are always dealt a hand of 13 tiles and you can go out on the 14th tile.

Mahjong gameplay goes counter-clockwise but you draw from the wall clockwise.

The Wall
  • Break wall using the values from 3 dice
  • Dead wall is 14 tiles long
    The seven tile stacks (2 tiles high) to the right of the wall break make up the dead wall. The dead wall continues around the corner to the next wall, if the end of the wall is reached. After the seventh tile stack, the two sections of the wall are pushed a bit apart to set the dead wall apart from the end of the 'live' wall.