A life points theory
Goal : to help resolving tsumegos and life and death situations in games.
Problem : it’s hard for new players to know how to make a shape in 6, 7, or more points dead or alive.
A theory that might help : hopefully reading this could help some players to “see better” the life points of any shape, and their impact on life and death.
Scope : This theory studies only inner-space of shapes, without outer liberties, cutting points, and not on border / corner situations. This will come later. So it can only be a part of the solution for tsumegos and life and death situations in games.
It’s quite easy to learn by heart the life points of shapes up to 5, since there are quite few :
shapes of size 1, 2, and 3
shapes of size 4
shapes of size 5
but there are 35 shapes of size 6, 107 shapes of size 7, 363 shapes of size 8, … it’s a lot to learn by heart.
Killing Shapes page teaches us that 3 shapes are dead, 5 shapes are conditional, and the rest is theoretically alive.
But things can go wrong :
- if defending player plays at the wrong point, the theoretically living shape can die
- if there are already some opponent’s stones inside, the shape can be dead, conditional, or alive. How to know that ?
- if the shape is definitely dead or alive, players should be aware of it, in order to tenuki instead of playing another stone inside
So here is rule #1: a shape of size x can be filled with 2 shapes of size x-1
the shape in 2 can be filled with 2 shapes in 1
the shapes in 3 can be filled with 2 shapes in 2
one shape is yellow, the other is blue, it is green where they overlap.
the shapes in 4 can be filled with 2 shapes in 3
Sometimes there are several possibilities
the shapes in 5 can be filled with 2 shapes in 4
Note : all rotations and symmetries also exist, but they were not drawn because there are too many and it changes nothing to the theory. That’s why in the pictures above, yellow shapes “don’t move” and blue shapes make every possibilities.
Actually there are 12 possibilities for the cross shape. But for each position of the yellow shape, the blue one can have 3. Let’s ignore the 9 symmetries.
And so what ?
And so, rule #2: life points are the ones that are always green
It means that life points are points where the 2 shapes of size x-1 always overlap :
for the T shape, life point is the one that is green in every variant
for the square, no point is always green
If you got the idea, here are the life points of shapes in 6 :
we start to see a pattern : rule #3 : the life points are before extremities
Imagine you want to go everywhere inside the shape : some points are the only way to reach extremities.
There may be more than 1 point before an extremity :
- first, let’s place a life point just before every extremity.
- Then we see that the bottom-right extremity and its life point can be reached only by passing through the bottom-center point, so it is a life point too.
- Point A is not life because you could skip it and pass by B.
- Point B is not life because you could skip it and pass by A.
another way of seeing it : rule #3b : the life points are where a x-1 shape has to pass
What points can you remove without splitting the x-1 shape ?
if you can think of a x-1 shape that doesn’t include a point, then that’s not a life point.
Look again at the first shape : can you “see” the life points ?
In case of doubt, use rule #1, #3, or #3b
Seeing life points helps us to fight :
rule #4 : if defender plays a stone on any life point, the shape lives
Indeed, life points are the ones that can split the inner space in two, thus making eyes.
That means that the attacking player has to prevent the defender to keep a stone on any life point. Not easy…
Reminder: This theory studies only inner-space of shapes, without outer liberties, cutting points, and not on border / corner situations. This will come later.
Does rule #4 mean that the shape dies when the attacker controls every life points ? in most cases, yes. but there is an exception :
the curious case of rectangle 6:
we can see that the rectangle 6 has no life point… but we know that this shape is theoretically alive ! Strange, isn’t it ? not really : since there is no life point, it means that the defender cannot make two eyes by playing just 1 stone. s/he has to “generate” life points.
It leads us to the rule #5 : if defender plays on a non-life point, the shape transforms
if defender plays on a non-life point, the shape of size x becomes a shape of size x-1 :
there are several reduction possibilities
but some have more life points than others
in most cases, it is useless since :
- either the life points of the new shape stay occupied by the attacking stones
- either 1 new life point is generated, and since it’s attacker’s turn, s/he plays on it
in one case, it is useful : rule #6 : rectangle 6 can generate several life points
The defending player can generate life points by playing on a corner, or a center. (the 4 other positions are just symmetries).
If he chooses center, it generates 3 new life points ! The defender can take one and live, unless the attacker already controls at least 2 of them like that :
even if the attacker controls every life point on a shape, if it includes a rectangle 6, the shape might get 2 eyes.
a rectangle 6 can fit in several shapes of size 7, 8, 9, or more.
reducing the shape by playing at the center of an inner rectangle 6 can generate a variable number of life points :
There are many possibilities, so as long as a rectangle 6 can fit in, both players should imagine them all. Shapes that contain many rectangle 6 are nearly impossible to kill since the defender just has to generate 2 unoccupied life points.
So the defending player wants :
- to play on a life point
- if not possible, and there is a rectangle 6 somewhere, then look if it can be reduced to generate 2 unoccupied life points.
- if not possible, then the shape is dead and players can tenuki.
And the attacking player wants :
- to occupy every life points.
- If there is a rectangle 6 somewhere, also prevent the defender to reduce the shape to an x-1 shape with 2 unoccupied life points.
rule #7 the shape dies when the attacker controls every life points and a rectangle 6 cannot fit in
If a shape is dead, both players can tenuki. The only interest for attacker to continue is to remove the stones from the board to help the outer situation.
That’s it. I hope this theory helped you a little. Next I would like to make another article for the same theory but including borders, corners, cutting points, and with outer liberties.