Crazyhouse chess is much more complex than regular chess. I wondered when chess already offers billions and trillions of different games that could potentially played how many options there are for crazyhouse games?
Is it even possible to calculate this mathematically? Does crazyhouse even have an end or does it go on forever? I mean as soon as the board is full with pieces there are no moves possible anymore at some point so there has to be an end, right?
If there is no end, how many games are possible till move 150?
That’s actually a very good question. I honestly don’t know
First of all, the crazyhouse board never has more than 32 pieces on it. so it will never get full.
For the finiteness of the number of games, the calculus is quite simple.
1) There is a finite number of fields on a chess board.
2) There is also a finite number of possible sets of figures (for each color). For example one of the players could have more than eight pawns on the board. But the number of combinations is finite.
3) A game ends, if the game enters the same state for the third time.
even without all the other rules of chess it can be prooved that the number of crazyhouse games is finite. so we will forget about those for now.
Let us think of a move as the transition of one position on a chess board to another one. Of cause a move can also change the set of pieces on the board (capture).
To find all the possible positions on the board with any possible set of figures, we would sum up the number of possible positions for all the possible sets of figures. As both numbers are finite, the sum is a finite sum of finite elements and therefore finite.
this means there is a finite number of possible states of the game.
Now thnk of the longest possible game under these circumstances. it would iterate twice though every possible state. and then the next move will put the game into a state it already was in two times. this ends the game.
So, every chasyhouse game does end.
Also there are finitely many possibilities to arrange any subset of a finite set. See a math book for a proof. This means that you can only have a finite number of games with a finite number of moves.
Now your last question is actually the hardest one. I have no idea how this could even be calculated efficiently. You cannot just play all of them. even if we did it on all our computers no one of us would live long enough to see the result.
You would need some smart way of estimating this number. I have no idea how to do this. I would just like to add:
for the complexity of the game it might be much more interesting to know how many interesting games there are. Maybe this can be guessed somehow from the games that are already known (see the lichess database)
@Tuedelmeister I wonder why your rating is so low despite your sharp mind and high intelligence. Thank you! I kind of forgot that you always have the same amount of pieces and the reason my opponent had 3 bishops was that one of them was initially my own. That makes sense now. So you cannot ever have 3 queens.
Very well analysed. I guess you can just more or less take the number of possible puzzle creations times two (+1 for the third repetition).
That would be 16 pawns, 4 knights, 4 bishops, 2 queens, 2 kings, 4 rooks put into random squares on the board, in every combination possible.
As we can also have less pieces on the board and pieces on the side, I guess it would be easiest to just count those places on the side as extra squares. So this makes a board of 64 + 1 = 65 squares.
Exceptions: The pawns are an exception, because they can only be put on 65 - 16 squares (without two ranks). So on 49 squares. The king is problematic, the only problematic piece in fact, as he cannot stand in a place where he is in check and the only way reaching it was via a place that also had been in check. The problem is there are positions with many illegal moves and lots of pieces and some with very few, so you can't apply anything here, but maybe it would help to assume that the king can approximately be on 64 (never on the side) - 10 places. This already makes the equation inaccurate, but I don't know a better way of dealing with it.
So the remaining question is how do we get the combinations out of this and as far as I remember this is possible with extrapolation or factorial. However I don't know how the formula would be. Having a formula would be kind of satisfying already. So maybe you can help me out there?
Thanks for your post.
@Ianalyse said in #4:
> @Tuedelmeister I wonder why your rating is so low despite your sharp mind and high intelligence
Being intelligent and knowing how to play a board game is not the same thing. There's hundreds of thousands if not millions of sharp people worldwide who couldn't tell you what backrank mate is
@WhatUpFam I disagree. After little time people figure out how to play a game better - intelligent people usually learn faster and therefore get a higher rating faster, no matter in what game. However I don't want to talk about this and I think it is pretty offtopic that you mentioned it here, because this discussion is about crazyhouse chess and not about a user, so I would kindly ask you to contribute to the topic or not and not make this offtopic. Thanks.
The main reason for my low rating is that i am not a very good chess player. Also I usually play in the evening before going to bed.
Keep in mind that there is no direct route from the number of possible board configurations (positions) to the number of games. If we assume that there are 100.000.000.000 possible positions (probably there are more), there are still only 20 possibilities for the first move and probably about 25 for the second move. this already makes 250.000 possible games after only two moves by each player and only one of those is called 'accepted queens gambit'.
Now for the number of games: It is relatively easy to calculate some number. the interesting part is to get a number you know something about. So In this case we would like to find an upper or lower bound for the number of possible games or at best get a very good estimate of the actual number of games. The easiest calculation would be to find out the average number of possible moves for any turn in a crazyhouse game. This number can easily be dominated by the pieces outside of the board, because they have at least 33 fields to be placed on (except for the pawns wich have at least 17). In contrast: A queen on an empty chess board has 27 or less fields to go to, a Rook has 14, a bishop has 13 or less, a knight has 8 or less, the king has 7 or less, and a pawn 3 or less (12 or less with conversion, 4 or less for ground row).
in practice the numbers for pieces on the board could easily be half or a third of what is stated above.
The other thing is the number of moves we want to look at. @Ianalyse, you picked a number of 150 moves. I wonder if you ever played a game with 150 moves and wether it was interesting. Currently there are 8.843.053 games of crazyhouse, 7.380 took 150 moves or longer. Many of them end in one player having 17 Queens all on the board. Only 637 games were started from the Lobby. That is 0.01%. I've read somewhere that the average chess game takes about 40 moves. to me this sounds like a more interesting goal to aim at.
Now one way to get a number of reasonable games would be to look at 100.000 random games of crazyhouse from the lichess database and find the usual number of moves. now for another 100.000 games of this length (there should be at least 2.000.000 with a good choice of length) one would average the number of possible moves for each turn and player over all games.
This number potionciated by the average number of moves in a game should give you an idea of the number of games to expect.
Note: I left out some important factors here:
1) There are situaltions in wich a plyer has only yery few possibilities to move, for example if he is checked. there are also situations in wich a player has exceptionally many possibilities to move, for example if he has captured one of each type of piece of hit opponent and is not in check.
2) There are many more possibilities to place a rook and a pwan on the board than to place twi rooks, because they are indistinguishable; id does not matter which one of the rooks is placed on a particular field because they are the same.
3) This approach neclects very short and very long games.
4) this approach does not take the rule of 3 into account (the game ends if the same position is seen 3 times. so to get an idea of the theoretical maximum of games one needs a more sophisticated approach.
At the Moment I cannot tell you the numbers. One way to get them is to take a computer and use the lichess API. I am not too much into this, I would need to look up how this works.
> Keep in mind that there is no direct route from the number of possible board configurations (positions) to the number of games.
Yeah you are probably right... The way of reaching all those positions is so complex that it wouldn't help that much to know how many there are, but I don't see another way of finding it out. It was still fun thinking about it. The way of counting 20 moves per move was tried for standard chess and turned out to be extremely inaccurate. I saw this on numberphile youtube.
> Many of them end in one player having 17 Queens all on the board. Only 637 games were started from the Lobby.
Wait what? I thought you can only have a fixed number of pieces??? You said before that there are always the same amount of pieces. Now I am confused!
> First of all, the crazyhouse board never has more than 32 pieces on it. so it will never get full.
The pawns can promote. When we consider pawn promotions it gets incredibly complex.
@Ianalyse said in #6:
> @WhatUpFam I disagree. After little time people figure out how to play a game better - intelligent people usually learn faster and therefore get a higher rating faster, no matter in what game. However I don't want to talk about this and I think it is pretty offtopic that you mentioned it here, because this discussion is about crazyhouse chess and not about a user, so I would kindly ask you to contribute to the topic or not and not make this offtopic. Thanks.
And I completely disagree with your take. Not every intelligent person that plays chess devotes their time to learning. There are many people who only play for fun and do not care nor take the time to improve. Even he said it, he plays right before going to bed meaning its just something casual. So you're wrong here. And this is a free country, I can talk about what I want. So I will kindly ask you to stop telling me what to do.