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At White's first turn there are $20$ possible moves: each pawn can move forward one or two spaces, or a knight can jump over the pawns to one of two positions each. Some moves are of course likelier than others (for example, in all the novice games I've seen, I've never seen anyone open with the king's pawn). Black also has $20$ possible moves at this stage.

At White's second turn, there is a small number of available moves based on what the first move was, plus a good player should take Black's first move into account. But what I want to know is, if we follow each of the $20$ scenarios created by the first move, count the number of available moves in each, no matter how strategically inadvisable the move may be (e.g., move the king forward) as long as it's valid by the rules of the game, delete duplicates (that is, count each move just once, not again and again in each branch that it's still possible to make it), how many moves are there?

Also, what is the most efficient way to tally up these moves? How far can we take this method, say, Black's second move, White's third, or further still?

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  • $\begingroup$ You are describing a breadth first search of the game tree. You do that and count up the moves. When you say delete duplicates, I don't think that is what you want. If White didn't play the e pawn on the first move you still want to count its two possible moves on the second. You may want to look for transpositions, so 1 a4, 2 e4 leads to the same position as 1 e4, 2 a4 $\endgroup$
    – Ross Millikan
    Feb 21, 2015 at 21:55
  • $\begingroup$ So you are asking how many legaly configuration the white player can have after his 2° turn? $\endgroup$
    – Davide F.
    Feb 21, 2015 at 21:55
  • $\begingroup$ @DavideF. Yes, that's what I'm asking about. $\endgroup$
    – user155234
    Feb 21, 2015 at 21:57
  • $\begingroup$ @RossMillikan If the first move is the king's pawn forward two spaces, it's possible for the second move to be the king's knight to the edge of the board. If the first move is the king's pawn forward only one space, the king's knight to the edge of the board is still a possible move, but I don't want to count it again. Though of course it does create a different board configuration for the rest of the game... $\endgroup$
    – user155234
    Feb 21, 2015 at 22:02
  • $\begingroup$ Just a side note: 1. e4 is not that uncommon. Wikipedia proclaims this is the most popular opening, but not as likely to secure a win for White as other openings (but this is from Wikipedia, so a grain of salt is in order). Of course to move the king forward in the next turn is a boneheaded move. $\endgroup$
    – Robert Soupe
    Feb 22, 2015 at 4:28

2 Answers 2

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Based on your comment, you are really counting the number of single moves possible in white's first two. Each pawn has three possiblities, one space, two spaces, and (if it was advanced two on the first move) from rank 4 to rank 5, for a total 24. Each knight has two for a first move and eight for a second move (including back where it started-is 1 Na3 2Nb1 different from 1Nc3 2Nb1? Does that count as four moves?), total 20. Each rook has two for a second move. Each bishop has seven. The queen has nine and the king three. Overall total $24+20+2\cdot 2 + 2 \cdot 7 + 9+3=74$ possible moves in white's first two. It depends also whether you think 2Bh6 is different from 2BxNh6 and 2Bxh6. I have counted it only once.

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  • $\begingroup$ For what it's worth, 20, 20, 74 does not give a chess-related result in an OEIS search. Maybe it's time for your second OEIS sequence? $\endgroup$
    – Robert Soupe
    Feb 22, 2015 at 4:35
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I just worked out how many possible ways a chess game can progress through White's second move. I got $8,902$, but I was outdoors and didn't use any paper so I may have made an error.

If we're just looking at White's second move and ignoring Black, I guess it would be... figuring backwards... it was $8,902$ and not $8,900$ because of the small number of moves Black can make that interfere with White, giving or subtracting a move. So, $8,900$ divided by $20$ is $445$. That's how many positions White can have after move $2$, but it's not clear whether that's exactly what you were asking. The poster above may have answered what you meant to ask.

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