Longest direct checkmate
It is always interesting to see the long checkmate positions. The more exciting is to compose a good problem yourself. One can easily compose checkmate in 2 or 3. May be 4. Composing mate in 5 will take time. But how long is the longest possible checkmate? (with the perfect play of both sides) Now we know the answer: 16 moves!
Longest checkmate line
We have seen direct mates in 16 - white moves first and checkmates, which means the line is 31 plies (half-moves) long. However, there is a position where it takes 32 plies until the checkmate - white to move, and survive for 16 moves. This position is unique in 3x3 chess - it is the only checkmate line of 32 plies. Of course, same position can be produced by mirroring left-right or swapping black and white (x4 more). And, since first move is forced Kb3, any black piece can be placed to b3, still giving same result (x6). It means, there are 24 positions with the same evaluation, but basically only one position.
Please, see that white has only King alone, but still can hold for 16 moves against 7 black pieces! The position is completely stuck by the Knight in the center, which has no hope to move or get captured. Black should carefully free the a1 square, then promote to Knight avoiding stalemate. Stalemate is one of biggest problems on such a limited space. Then black should give away his Queen and Knight, to be able to move pawn c3. After a while black promotes that pawn to queen (or rook), and gives it away to be able to checkmate finally. The main black's trouble in this line is to reduce the number of his own pieces.
Longest line with limited number of pieces
You might be curious about longest checkmate line with limited number of pieces on board. It is interesting if you consider parallel with chess, where only up to 5-men endgame is completely solved currently. Here I will show the longest lines for limited number of pieces on 3x3 board. (It means - longest line where both sides make only best moves).
Most movable positions
Here we will look for positions with as many possible moves as possible. For example, in this position (two kings in the corners) white has two moves - Ka2 and Kb1. There are positions with more choices to move, like this one - 7 different moves. But which is the most movable position? This question is not difficult, since you don't need to consider the outcome of the position. Just shuffle the pieces on board. When you are ready, check the answer (or another one).
Most 'unhappy' position
Most 'pointless' position
Most 'happy' position
Best move problems
Here we will look for difficult positions, where there are many moves, but only one decides the game. First, the champion position: only 1 winning move out of 17. Next, the position with ony one winning move out of 16, where other 15 moves are all draws.
Now let's see some very dramatic positions. Only one move is winning, other moves are all lost. First one is mate in 5, with 13 possible losing moves. As you see, the winning move is promotion to knight. This is the common theme in interesting problems (1, 2, 3).
Now let's find the only draw in almost lost position. Here white has 16 moves to choose from but only one move can save him. Consider, that some of the alternatives are losing in 8 or 10, which makes it difficult for human to find the correct move. Again, underpromotion is required, as well as in this similar example. But underpromotion is not the rule, here we have position with usual healthy promotion to queen.
Full-point mutual zugzwang
... is a position which is lost for the side to move in case of both sides. It is won by black if it is a white's turn to move, and by white otherwise.
Although in chess stalemate is just ordinary draw, in this study to stalemate is considered better than to be stalemated. This is influenced by shogi, where stalemate is winning. With this assumption, although stalemate is still draw, it is possible to have 'distance to stalemate' measure in the same database, without any conflict with usual 'distance to the checkmate' information. OK, so stalemate is still draw, just I assume a player will prefer to make stalemate, rather than to be stalemated.
Now, let's see the longest direct stalemate of 3x3 chess - stalemate in 14. There are totally 92 such positions, with only five basic ones. Other are just left/right symmetric, side swapped, or having different piece taken on the first move. Here are the basic ones: