With further testing I've found that for the c6 and Nf7 positions only 1. e4 is good for White, the others are all better for Black, the c2 position has 1. e4 and 1. d4, while the two equal positions, Nb7 and Nc1, have 1. e4 and 1. Nf3.
An Equal Opening Position in Crazyhouse?
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I've also determined another quite equal position. In contrast to the previous ones, which are either equal or slightly favoring Black, this one favors White. It has an extra Black pawn at b6. Its eval is quite volatile, ranging around between 0 and +1 in my tests, so at first I discarded it. However, now I will include it back since its average eval is +0.5. White's slight advantage is similar to the one in standard chess. The position's best move is 1. e4, and 1. d4 works too.
(Also as an honorable mention of some sort is an interesting position in which Black has a bishop instead of a knight on b8. This starts out seemingly better for White in the eval, but then it drops to a very variable range between -0.4 and -1.3. Average is about -0.6 to -0.8. I include it here because I think it is very interesting for a player to have a "bishop triplet".)
According to the Lichess Stockfish Crazyhouse Engine, 1. e4 is a whopping +2.9 (depth 39 on Lichess). Black's best response is 1... Nf6 2. e5 Nc6!, with 3. exf6? instantly equalizing after 3... gxf6 and creating a solid position for black. This is known as the Crosky Gambit, and is Black's best chance to counterattack White. However, White can simply decline this gambit with 3. d4, after which there is 3... Ne4 4. f3 d6 5. e6! Nf6 6. exf7+ Kxf7, with a winning attack for White.
So, I've been thinking about slightly altering the starting position to make the game equal. Here is what I've found using a depth 24 Stockfish engine on Lichess and only altering one square on the board (note that I don't claim to have discovered any of this; it is very likely that some else has done so already):
3-way tie for 2nd place, eval -0.5:
- Black with an extra pawn on c6. This gives Black an extra pawn but blocks Nc6.
- White with a missing c2 pawn. In contrast to losing a central pawn, which actually isn't that big of a deal, losing this c pawn is more of a challenge for White. However, losing the f2 pawn is even worse and is quite good for Black.
- Black with a knight on f7 instead of a pawn. Black has a knight controlling the center instead of a pawn, but this weakens Black's kingside diagonal.
2-way tie for 1st place, eval 0.0:
- White with a knight on c1 instead of a bishop. This is quite an unbalanced game from the perspective of the minor pieces, but since White loses a bishop that could have been key in controlling diagonals, the game is equal. Interestingly enough, knights are supposedly worth 3.5 and bishops only 3, but I guess the engine is smarter than us humans when it comes to evaluating these small "advantages".
- Black with a knight on b7 instead of a pawn. Again, Black has a knight for a pawn, but it is more on the flank this time. Black may plan to fianchetto on the queenside after the knight moves. Since both sides have an equal number of bishops, I think this is more preferable to the previous one.
The evals with probably be different with higher or lower depth, but they will maintain overall consistency.
What do you think? Do you enjoy crazyhouse? (If you're like me and totally cannot see 24 moves into the future, crazyhouse is more like this times 100: why not, let's try, who cares about blunders; sacrifice, sacrifice, sacrifice, oh no, defend, defend, yes, sacrifice...!)