Tile Theory at Two-Shanten (1)
Let us begin with the pattern where you compare floating tiles.
This applies to hands that already have one completed set, one completed head, and three taatsu blocks including pair blocks.
Of course you should not break the completed set, and if there is only one head, breaking that is also forbidden because it slows the hand sharply.
Either choice sends the hand back to three-shanten.
So the basic rule is to list the floating tiles as candidates first, and compare among them.
Example 1
Tsumo
=> Cut
The manzu block
already forms one taatsu block here.
If you touch a taatsu block, the acceptance toward one-shanten decreases.
So the basic rule is to choose from the floating tiles.
That leaves a simple comparison between and , so naturally the correct discard is .
Example 2
Tsumo 
=> Cut
Here you compare the values of and , because neither of them is part of a taatsu block.
The manzu side is already stronger just because it forms a connected shape.
As for souzu, because there is already a ryanmen taatsu in
,
drawing 
takes the hand directly into one-shanten, so there is almost no point in keeping the floating tile.
At most it leaves a tiny Sanshoku possibility in 345.
If you draw 
, you end up with a nido-uke situation against
,
so it is still worse than the manzu side.
Example 3
Tsumo 
=> Cut
Here you compare the values of and , since neither belongs to a taatsu block.
Even though the souzu side looks connected in
,
its ability to make sets is still far inferior to that of an inside middle tile.
Example 4
Tsumo
=> Cut
This is quite a delicate choice.
Because there are many pairs, the nobetan value of
-
is low here. So from a pure tile-theory viewpoint, the difference between
and is actually very small.
That means this is where you take the difference in yaku value.
Even if the pinzu side becomes a ryanmen taatsu, it does not move toward Tanyao.
But on the souzu side, draws from to can all still bring Tanyao along.
Up to this point, every example has chosen from floating tiles.
So are there really hands where you should touch a taatsu block instead? Yes.
Example 5
Tsumo
=> Cut
In a three-pair shape like Example 5, there is no need to insist on keeping the shanpon acceptance.
The floating tiles are
,
but the better play is to keep both of them and cut instead.
Rather than clinging to the loss of one tile of current acceptance, it is better to preserve the chance to improve into better shapes.
There are also hands where you should deliberately keep the less functional tile.
Example 6
Tsumo 
=> Cut
This applies to hands like Example 6, where the existing taatsu are already all nicely arranged as ryanmen blocks.
Even if you keep , it can never grow into a taatsu stronger than the ones you already have.
In that case, may well become a dangerous tile later, so it should be discarded now.
Theory
Original Japanese page: http://beginners.biz/pairi/pairi10.html