Tile Theory in Tenpai
When you are in tenpai, there are often no choices at all.
But if you can choose your wait, the basic rule is to choose the one that is easiest to win with.
1. Number of Winning Tiles
The more winning tiles a wait has, the easier it is to win.
So naturally, you begin by choosing the option with more tiles.
Example 1
Tsumo 
Pon 
Example 1 offers four different wait choices.
| Discard | Winning tiles | Count |
|---|---|---|
|
|
3 |
|
 |
6 |
|
|
4 |
|
 |
10 |
If you simply list them all, it becomes obvious that discarding 
is correct.
However, players who are not used to multi-sided waits often take the easy shanpon by habit.
This is not some rare sanmenchan at all.
Once you know the shape, you should be able to make this choice instantly.
Example 2
Tsumo 
For safety, let me say this clearly: even if a wait has more types, that means nothing if the actual tile count is low.
Example 2 is a case where you should not take the sanmenchan.
| Discard | Winning tiles | Count |
|---|---|---|
|
|
5 |
|
|
6 |
Even though one choice looks like a three-sided wait, the actual number of tiles is small, so it is worse than a nobetan.
And because it would also break the Iipeikou that is already likely to score,
Example 2 should riichi on the 
wait.
2. Comparing Improvements
If the number of winning tiles is the same, compare how easily each wait can improve into a better one.
Example 3
Pon 
Kanchan or shanpon?
This is a common choice, and both waits currently have four tiles.
So do not rely on feeling. You should always compare the number of good-shape improvements.
Example 3 is a hand for open tanyao, so even if a ryanmen appears, any shape that wins on only one side should be rejected.
That means 
and 
are not really effective tiles.
The truly effective improvements are:
If you take the shanpon: 
If you take the kanchan: only
So here, taking the shanpon is clearly best.
Example 4
Tsumo Chi 
At first glance this looks like a perfectly even choice between two ryan-kan shapes, but the basic rule is still to take the kan side.
Why? Because the effective tiles are 
and respectively, and can be called from kamicha.
Chi
→ Chi 
Chi
This is a slightly more advanced way of thinking, because it includes not only self-drawn improvement but also the ease of improving by calling.
There is also a simpler kind of judgment: use the live table.
Suppose a ryan-kan shape is 
.
If, for example, three copies of 
are already visible in the discards, then even if you draw 
and turn it into an 
wait, that result has little value.
In situations like that, it is important to catch this sort of table information and respond by cutting , and so on.
The comparison between tanki and kanchan or penchan needs special care.
In these cases, you should often take the tanki even if it has one fewer tile at the moment.
Example 5
Tsumo
Penchan has four tiles, while tanki has three.
But that does not mean you should thoughtlessly riichi on the penchan.
This is the kind of hand where you should stay dama on the tanki and wait for improvements, aiming to riichi only after the wait becomes at least ryanmen.
This is especially true when the hand contains a concealed middle-tile triplet or connected sequences, because the number of improvements becomes very large.
In Example 5, drawing any of the following gives you a good-shaped riichi:
So in this case, it is completely reasonable to say that postponing riichi has plenty of value.
Original Japanese page: http://beginners.biz/pairi/pairi19.html