The next category to learn is Triplets and Kong. (Since kong is a Chinese word, its plural is also kong.) Here are the patterns in this category:
4.0 Triplets and Kong
4.1 All Triplets : 30
4.2.1 Two Concealed Triplets : 5
4.2.2 Three Concealed Triplets : 30
4.2.3 Four Concealed Triplets : 125
4.3.1 One Kong : 5
4.3.2 Two Kong : 20
4.3.3 Three Kong : 120
4.3.4 Four Kong : 480
Collecting triplets (pungs) and kong is much more difficult than collecting sequences (chows) - getting one triplet set is about ten times more scarce than getting one sequence. Collecting concealed triplets is even more challenging. Chinese Classical mahjong doubled the score of each triplet if it was concealed. That's a great start, but it turns out collecting several concealed triplets is so rare that it should be rewarded when it happens.
These patterns combine with others nicely: One Suit and Honors, One Suit Only, All Terminals and Honors, All Terminals, All Honors, sets of Dragons, sets of Winds, Consecutive Triplets, Similar Triplets.
4.1 All Triplets
This category starts with a medium scoring pattern. Any hand containing all triplets (no sequences or chows) scores 30 points. This is a fine pattern to attempt when you start a hand with (or find yourself suddenly with) four or five pairs. Only about once in 8 games will you be able to complete this pattern. You need those pairs to start with so you can claim pung during the hand. If you have that many pairs to start with, you might also consider Seven Pairs, which also scores 30 points, but you have to get at least 13 of those tiles yourself. With this pattern, you could start with nine.
This is a difficult hand to defend against, because the sets need not be related to each other. If you've shown a triplet of 2 dots, and another of 9 characters, that doesn't tell me anything about the other sets you're trying to complete.
4.2.x Concealed Triplets
If you want to keep your options open, and try for concealed sets or perhaps the Seven Pairs pattern, pass on the first one or two tiles you can claim early in the hand. Getting two or three concealed triplets will help your score nicely. Getting four of them will happen about only once in 400 hands, but it does happen, and so it is rewarded with a handsome score. Other styles do give a hefty score for this pattern. These patterns might be considered as having too low a value. I think that is a conscious decision by Alan Kwan to encourage claiming of tiles, which makes for a more exciting game.
Showing a triplet set lets everyone know where those three tiles are, so be careful claiming early in the hand.
4.3.x Kong
Kong sets are exciting to get and to witness, and they twist the flow of tiles for a moment, which is a nice change-up. Getting four kong in many mahjong styles scores the agreed-to limit, and it is indeed rare - maybe one hand in 10,000. On the other hand, three kong is a much more attainable pattern (perhaps one hand in 500).
In this category, you have to balance the ease of claiming a tile for pung versus hoping to draw a third tile from the wall to get a concealed set. You also have to remember that if you do pass on a tile, you cannot get a kong with that tile later. Of course, sometimes you will have four of a tile in your hand, and you will not want to create a kong because you make a triplet and a sequence with those tiles.
If you do find yourself with four identical tiles in your hand, especially early in the deal, consider a half-flush or full flush. You can wait to call kong on any turn you start with a draw from the wall, so keep your hand flexible if you can at the start. On the one hand, you get the extra tile when you declare the kong, which could be a big help. On the other hand, keeping your opponents guessing can be valuable, and if you are ready to go out, that replacement tile could be your winner and give you a few extra points.
Declaring a kong early in the game can make players change strategy. If you show a kong of 7 bamboo, that crushes my hopes for a nine-tile straight, several three similar chow patterns, or even some consecutive pungs. If you show a kong (or even a pung) of honor tiles, that stops my hopes for thirteen orphans or maybe big three dragons. If you do it late in the hand, there may be nothing I can do to recover.