Smallest Groups with Two Eyes
Today’s riddle is for the Go player. Your task is to find all the smallest groups with two eyes and place them all together (with the corresponding enclosing enemy stones) in a single board. Let me give you some tips first:
- Smallest groups in the corner: In the corner, six stones are the minimum needed to complete any group with two eyes. There are only four possibilities, and I took the liberty of placing them on the board for you:
- Smallest groups on the side: Consider any of the smallest groups with two eyes on a side of the board. How many stones do they have? [Hint: they all have the same number of stones] How many different groups are there?
- Smallest groups in the interior: Consider finally any of the smallest groups with two eyes in the interior of the board. How many stones do they have? [again, they all have the same number of stones]. How many different groups are there?
Since it is actually possible to place all those groups in the same board, this will help you figure out how many of each kind there are. Also, once finished, assume the board was obtained after a proper finished game (with no captures): What is the score?
Although these riddles are purely combinatorial (and thus no great deal of expertise in go is needed), I would like to take the opportunity to encourage everyone to learn this game: so simple, yet so complex, and richer in strategy and tactics than any other game I have ever seen before—including chess! The appealing for the mathematician is obvious, if only for the possibilities in combinatorics that arise from the mere setup of the game. There are many good series of books on the subject, but I found the following specially well written, and very accessible to players at every level. Enjoy!