To succeed in Mahjong, you first have to learn how to upgrade your hand to the point where you can actually win. Getting to tenpai (ready state) as quickly as possible, while not always the best idea, is an important skill to have.
A good way to look at it is car racing, or speed skating. If you are ‘faster’ than everyone else at the table, nothing else really matters – you’ll get to Tenpai before everyone else much more often. However, if everyone is equally ‘fast’, then a lot more factors come into play, and you can even fall back and wait for everyone to trip over each other and cruise to victory.
There are two important terms that come into play here: Shan-ten and Uke-ire. Shan-ten is ‘draws to tenpai’. This is the minimum number of different times you’d have to change your hand to get to tenpai. Think of it as the “number of perfect draws needed”. Lower is better. Zero is Tenpai. Uke-ire is ‘total possible good draws’. This is the total count of tiles that, if drawn, allow you to lower your Shan-ten.
To get close to tenpai, Lower your Shan-ten – and if you can’t, raise your Uke-ire*.
(*in some rare cases, you can greatly raise your Uke-ire by raising your Shan-ten – so much that your hand actually becomes faster. Like taking one step back so you can take two or three steps forward more quickly, That’s for later, though, under “Edge Cases” below.)
Example: Counting Shan-ten and Uke-ire
Let’s look at a pretty close hand.
A hand is four sets of three and a pair. Looking at the hand, here’s the ‘pieces’ of it:
: this is 2/3 of a set, waiting on a 
.
: this is a pair (which can also be 2/3 of a set).
: this is 2/3 of a set, waiting on a 
or 
.
: this is 2/3 of a set, waiting on a 
or 
.
: this single tile is 1/3 of a set (or 1/2 a pair).
: this is a set.
: this is another single tile, 1/3 of a set (or 1/2 a pair).
So, you have 1 set and 1 pair already, and 3 more sets that are 1 tile away. Since 2 changes would get you to tenpai (the 3rd would complete your hand), your Shan-ten is 2. The tiles that get you closer – lowering your Shan-ten – are ,,,,, and there’s 4 of each in the deck, so your Uke-ire is 20.
Now, there are certain tiles that can raise your Uke-ire, even though they don’t lower your Shan-ten. In this example, a 
. If you drew that, and discarded, say, the , you are now also waiting on a 
or 
– that’s 8 more Uke-ire.
Or if you drew a 
, and discarded the . Now instead of just waiting on a , you are waiting on and 
– 4 more Uke-ire.
You can use this to determine what to discard if you have to pick between multiple things. For example:
Using Uke-ire to determine what to discard, #1
and you draw a .
There’s 6 parts to this hand:
– set
– set
– pair
– 2/3rds of a set
– 2/3rds of a set
– 2/3rds of a set
You have to get rid of something, so your choices mean you have to break up one of these six parts. and are complete sets, so no. 11s is your pair, so no. now you have to get rid of , , or . is waiting on two different tiles compared to the other ones, so that’s out.
Now, to pick between and . while both are only waiting on one of the four copies of one tile, for the there’s one tile () that lets you then wait on two tiles instead of one ( waiting on 
), whereas there are two tiles () that do the same for the ( or ). So, losing the 13m is best. between the and the , the is better to lose, as there is a tile () that still gets you that preferred two-tile wait.
Using Uke-ire to determine what to discard, #2
draw 
(this is a pair, waiting on 6 total tiles to become a set – the 4 s, and the 2 s)
the worst parts of your hand are the 1-tile parts. the and are only improved by s and s (3 uke-ire), the with (19 uke-ire), the works with , 
and the 3 remaining (11 uke-ire), the with . To note, though is that the is actually “worse” than the , as the first and drawn are used by the , so the is really only a 9 uke-ire.
Of course, this guide is only to help you find the fastest way to get closer to tenpai. There’s all sorts of reasons to make the ‘less efficient’ discard – defense, laying traps, pushing for a higher-scoring hand – but knowing the fundamental rules is the first step in learning when to break them.
For instance, there are times when increasing Uke-ire can be worth it, even at the price of getting farther from tenpai. Edge waits are the good example.
Edge Waits
Something to note is that penchan (Edge waits, like or 
) are really hard to complete – not only is there only one tile, but you have to upgrade the wait twice > > before it gets any better.
Single tiles with lots of room around them (for example, ), while two tiles away from becoming a set, have lots of ways to get better – they can easily become multi-tile waits (ryanmen, such as ), or at the very least center waits () that upgrade easily to multi-tile waits. In fact, with this hand:
Choosing between discarding vs. or , the difference in number of draws between which sequence completes first is less than 1%.
So, if you have a number of unconnected middle tiles, maybe keeping them all and seeing which one develops is a better idea than keeping that penchan. Try it yourself!