And if White can win how should they play?
To get answers to these questions, it is enough to write a simple program that analyzes all possible variants (like ChessExplorer), enter the initial position on the chessboard, search for a checkmate, e.g. in 50 moves and ... wait long enough.
The waiting time for the result would indeed be quite long, because the number of variants to be analyzed, as it turns out, is huge.
So let's try to estimate the number of possible variants in a chess game.
At the beginning of the game, White has 20 possibilities, in the next move Black also has 20 possibilities, i.e. in the first two single moves there are 400 different variants of the game.
The number of possible continuations increases rapidly with subsequent moves:
| in 3 first moves | - | 8,902 | |
| in 4 first moves | - | 197,281 | |
| in 5 first moves | - | 4,865,609 | |
| in 6 first moves | - | 119,060,324 | |
| in 7 first moves | - | 3,195,901,860 | |
| in 8 first moves | - | 91,768,468,861 |
Of the latter number, 8,102,108,221 are variants starting with the move 1. e2 - e4 and only 2,863,411,653 with 1. a2 - a3.
Analysing this above numbers you may notice that White is here a little privileged. In 3-th move (White) there is an average of 22.26 possible replies (8902:400), whereas in 4-th move (Black) - 24.16, in 5-th (White) - 24.66, in 6-th (Black) - 24.47, in 7-th (White) - 26.85 and in 8-th (Black) - 26.60.
As the game develops, the average number of possible moves continues to increase, but in a different way depending on the opening used:
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Sicilian Game 1. e2 - e4, c7 - c5 2. Nf1 - f3, Nb8 -c6 3. d2 - d4, c5 x d4 In this position, the number of further possible continuations of the game is:
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King's Gambit 1. e2 - e4, e7 - e5 2. f2 - f4, d7 - d5 3. e x d5, e5 - e4 Number of possible continuations:
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King's Indian Defense 1. d2 - d4, Ng8 - f6 2. c2 - c4, g7 - g6 3. Nb1 - c3, Bf8 - g7 Number of possible continuations:
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Nimzo-Indian Defense 1. d2 - d4, Ng8 - f6 2. c2 - c4, e7 - e6 3. Nb1 - c3, Bf8 - b4 Number of possible continuations:
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It is visible here, for example, that in the King's Indian Defense Black has less room for maneuver than White, and in the King's Gambit - the opposite. In this last sharp debut, the number of possible combinations is increasing faster than the average in other variants.
In the middle game, the number of possible continuations varies understandably depending on the position, ranging from about 20 to about 50 possible answers per move.
In the endgame, the number of pieces on the board is usually smaller, and similarly the number of possible variants is smaller.
Let's now to count complete amount of chess variants.
- In first 4 full moves is 92 billion of variants, that is about 1011.
- Let's assume that in 26 moves (from 5. to 30.) in each is 1000 variants (about 32 in single move).
- In 5 moves (31. - 35.) - 500 variants per full move.
- In 5 moves (36. - 40.) - 200 variants per full move.
- In 10 moves (41. - 50.) - 100 variants per full move.
If a computer will analyse 3,000,000 variants per second, we will receive the result after ... 10120 years !
Yes! We have quantity of possible chess games: 10134 .
And second question: what's quantity of possible positions on chessboard?
This problem is exactly defined but difficult to solution. We can try to dissolve this following way.
For every possible quantity of pieces (from 2 to 32):
- we count number of possible, different sets of pieces
- we take representative set of pieces and we count quantity of possible arrangements on chessboard (they should here to be only legal and possible positions).
For example, with two pieces we have only one set: two Kings.
With 3 pieces it exists 10 different sets (two Kings and one from 10 other pieces). With 4 pieces - 55 sets (they can here to be two Queens of one colour also).
For 16 pieces it exists ca 809,000 sets and for 24 pieces - ca 13,000,000. For 32 pieces we have only one set (promotion of pawn is impossible without capture).
Difficult is counting number of possible arrangements for given set of pieces. Unfortunatelly, this numbers are much larger and this they decide about final result).
Two pieces (white&black Kings) we can put on chessboard onto 3,612 ways (they can not checks themselves).
For 14 pieces (I put here 2 Kings, 1 white Queen, 2 white Rooks, 3 white Pawns, 1 black Queen, 1 black Rook, 3 black Pawns) I got 6.1021 possible arrangements.
I received largest quantities ( from 1029 to 1033 ) for numbers of pieces from 22 to 25.
Above numbers are already corrected with coefficients for eliminate illegal positions.
Final result - total quantity of possible positions on chessboard - accord to my calculations, he hesitated from 1038 to 1041.
I think it is here to accepting number 1040.
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