Life and Death: Go
It would be hard to imagine a much simpler game than Go. The idea, rules and general idea can be explained in three or four sentences. The technicalities and details might take another couple of paragraphs. I know of no game deeper than this three or four thousand year old one though. For those who would pursue mastery, or even basic competence, there are many technical details to master, and probably none is more basic than the matter of Life and Death.
The objective of the game is to surround territory, and it usually happens that groups of stones will be surrounded by stones of the opponent. In this situation, the stones are safe (alive) if they can remain in contact with two separated open points, but are captured (dead) if they can't. Assessing whether a given group of stones is alive or not is consequently a crucial game skill. If you or your opponent has an endangered group, it is essential that you be able to assess whether it can be killed or not. Killing a big group of your opponents stones is often decisive. It's also important not to waste precious moves saving the already safe or trying to save the already dead.
There are at least two crucial elements in skill at Life and Death. The first, and more basic, is pattern recognition: its crucial to recognize the status of key formations and their variants. The second, which is nearly useless without the first, is calculational ability: the ability to mentally work through the possible variations to see the outcome. Computers, of course, are great at the second but not the first. I am modestly proficient at the first (at the weaker club player level) and lousy at the second. In practice, that means that I can often solve problems that are alike enough to ones I have solved before.
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