Enumerating all 3-ply openings

Well, there are 20 openings in 1 ply, 400 = 20² openings in 2 plies and > 8000 = 20³ in 3 plies. Let's count it exactly by enumerating the number of possible moves for all 400 positions in 2 plies.

N.B. After that we will also count the number of possible positions in 3 plies = number of openings modulo transpositions (cf. posts #9-10). 😎

• 1. e4 b5 Polish Gambit*: 29 moves
• 1. e4 d5 Scandinavian Defense: 31 moves.
• 1. e4 e5 KPG: 29 moves
• 1. e4 f5 Duras Gambit: 31 moves
• 1. e4 xx Other Cases: 30 moves.

So there are 16x30+2x31+2x29 = 600 KP openings (starting with 1. e4) in 3 plies. 😎

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(thanks to @Ilampozhil25)

• 1. d4 c5 Old Benoni: 29 moves.
• 1. d4 d5 QPG: 27 moves
• 1. d4 e5 Englund Gambit: 29 moves
• 1. d4 g5 Borg Gambit: 27 moves
• 1. d4 xx Other Cases: 28 moves.

So there are 16x28+2x29+2x27 = 560 QP openings (starting with 1. d4) in 3 plies.

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(thanks to @PunchboxNET , cf. posts #14-18)

There are:

22x17 + 23x2 + 21= 374 + 62 + 21 = 457 moves for 1.c4.

17x20 + 2x21 + 19 = 401 moves for 1.f4

17x21 + 22x2 + 20 = 357 + 66 + 20 = 443 moves for 1. b4 and 1. g4

18x21 + 22 + 20 = 378 + 42= 420 moves after 1. a4 and 1. h4

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Total number of moves for 1.a4 - 1.h4:

600+560+457+401+2x443+2x420 = 3744

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Total number of moves for 1.a3 - 1.h3: = 3598. (thanks to @PunchboxNET , cf. post #28)

(to be continued)

Counting example (Sicilian Defense):

• 15 pawn moves
• 5 knight moves
• 5 bishop moves
• 4 queen moves
• 1 king move

15+5+5+4+1 = 30. This number can be called degree of freedom (of a given position).

true

so after e4, and a black move, white can do 601 moves(in total)

1.d4 e5 29

1.d4 c5 29

1.d4 xx 28

hey, this seems easy to do

oh idk why it was deleted but 1.d4 d5 27

Good! , so we have 28x17+2x29+27 = 561 openings in 3 plies starting with 1. d4.

The difference between 1. d4 and 1. e4 is that we have 2 vertical queen moves instead of 4 diagonal queen moves.

Where did 20 come from?

Edit: possible 1st moves?

Exactly.

At moves 2 you need to address transpositions, if you're counting openings.

Example:

1.e4 e5 2.Nc3 Nf6 Vienna Falkbeer

1.Nc3 e5 2.e4 Nf6 Vienna Falkbeer from Dunst

Well, indeed, let's distinguish chess openings (in the pure sense) and chess positions. 2 non-identical chess openings are considered to be different even if they give the same position. Later we could calculate the number of chess positions = number of chess openings up to transpositions. 😎

This must be more about math than chess then.

so this is how they figured out how many possible positions there were

Sure (up to 4 plies), however it becomes more complicated at 5+ plies. 😎

c4 xx = 22

c4 b5= 23

c4 d5 = 23

c4 c5 = 21

so 22x17 + 23x2 + 21= 374 + 62 + 21 = 457 moves for 1.c4

f4 xx = 20 moves

f4 g5 = 21 moves

f4 e5 = 21 moves

f4 f5 = 19 moves

so 17x20 + 2x21 + 19 = 401 moves for 1.f4

b4 xx = 21 moves

b4 a5 = 22 moves

b4 c5 = 22 moves 

b4 b5 = 20 moves

so 17x21 + 22x2 + 20 = 357 + 66 + 20 = 443 moves for 1. b4

g4 xx = 21 moves

g4 h5 = 22 moves

g4 f5 = 22 moves 

g4 g5 = 20 moves

so 17x21 + 22x2 + 20 = 357 + 66 + 20 = 443 moves for 1. g4

a4 xx =   21 moves

a4 b5 =  22 moves

a4 a5 =  20 moves

so 18x21 + 22 + 20 = 378 + 42= 420 moves after 1. a4

h4 xx =   21 moves

h4 g5 =  22 moves

h4 h5 =  20 moves

so 18x21 + 22 + 20 = 378 + 42= 420 moves after 1. h4

Wow thanks for these calculations! I'll add to the first post asap.

Total number of moves for 1.a4 - 1.h4:

600+560+457+401+2x443+2x420 = 3744