Welcome our new Croatian players!

A group of chess enthusiasts from Croatia have recently joined our ranks. Let's all extend a warm welcome to our new friends from Eastern Europe!

Look like that gothic chess, just like standard chess, soon will be dominated by Eastern Europeans. This is the way it should be.

£ukasz Maria P. Pastuszczak, also Eastern European.

I also welcome the new players.

Hello everybody!

Thanks for a warm welcome!

It's really a small group, but soon I guess, Goth Chess will be very popular here.

A great game with zillions of new possibilities and combinations!

See you online!

Well, "zillions" could actually refer to something else, so let's see if we can figure out how many Gothic Chess positions there are.

It would be one heck of a project to get the exact answer, but we can set a reasonable upper bound.

Let's start with the kings first. We could say the first king can go on any of 80 squares, leaving 79 for the other king, but this is getting off track already. Kings cannot be adjacent to one other.

You could even take adjacency into account, and still run into a problem with you start counting the pawns. For example, if you let the kings run all over the board first, sometimes a king will occupy a square slated for the pawns, and sometimes it won't. If a king is occupying a square that a pawn cannot occupy, there are more squares available for the pawns then you otherwise might count. If a king is occupying a square that a pawn might be able to occupy, there are fewer squares avalable for the pawns.

See how this is getting complicated?

To get our estimate accurate from the beginning, we need to break our calculation into several smaller calculations.

1. Both kings on rank 8 (no interference with any pawns). 2. One king on rank 8, the other king on rank 1 (no interference with any pawns). 3. Swap the kings from rank 8 to 1, and recount this number (no interference with any pawns). 4. Both kings on rank 1 (no interference with any pawns). 5. One king on rank 8, the other king on ranks 2-7 (one slot unavailable for pawns). 6. One king on rank 1, the other king on ranks 2-7 (one slot unavailable for pawns). 7. Swap the kings from rank 8 to 1, and recount those numbers (one slot unavailable for pawns). 8. Both kings within ranks 2-7 (two slots unavailable for pawns).

Do that for all situations for 0 to 20 pawns, and you have yourself the subset of king and pawn endgame counts only!

That was the easy part! Now you had to gradually start adding pieces, one at a time, and do the same types of caclulations!

You can see it would be nearly impossible to telescope this calculation forward, so let's try a different, less accurate approach.

1. Place 10 white pawns on the board first.

That is easy to count! The first pawn can be on any one of 60 squares, since it cannot be on ranks 1 or 8. The 10 squares on rank 1 are behind where the pawns start, and should a pawn get to rank 8, it gets promoted and is no longer a pawn. The next pawn can be on any of the remaining 59 squares, the next one, 58 squares, etc. So you get 60 x 59 x 58 x 57 x 56 x 55 x 54 x 53 x 52 x 51 which is equal to 273,589,847,231,500,800 (273 quadrillion, 589 trillion, 847 billion, 231 million, 500 thousand, 8 hundred!) But wait, fortunately, there are many ways to arrange these pawns that are essentially duplicates. This number is 10! which is 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 or 3,628,800. So, dividing 273,589,847,231,500,800 by 3,628,800 gives a very reasonable 75,394,027,566.

In mathematical notation, we say this is "60 choose 10" or 60 x 59 x 58... until we have 10 terms being muliplied, then we divide by 10 factorial, which is 10 x 9 x 8 ... x 1.

60 choose 10 = 75,394,027,566 ways to place 10 white pawns on the board (over 75 billion, or 12.5 times the population of the earth!)

Now, of course, many of these positions are not reachable. For example, pawns on squares a2,a3,a4,a5,a6,a7,b2,b3,b4 and b5 would be impossible, but it is still included in the count. The point is just to enumerate the arrangements to get an appreciation for the "depth" of the game.

Onward!

2. Place 10 black pawns on the board next.

50 choose 10 = 10,272,278,170 ways to place 10 black pawns on the board if 10 white pawns are already on the board. Each one of these positions can be reached from each of the positions with the 10 white pawns on the board, so the total number of ways to arrange 10 white pawns and 10 black pawns on the Gothic Chess board is 774,468,423,514,600,034,220 (774 quintillion, 468 quadrillion, 423 trillion, etc...)

3. Place 2 white rooks on the board next.

60 choose 2 = 1,770 ways to place 2 white rooks on the remaining squares. Multiplying this by 774,468,423,514,600,034,220 brings the total so far to 1,370,809,109,620,842,060,569,400.

4. Place 2 black rooks on the board next.

58 choose 2 = 1,653 ways to place 2 black rooks on the remaining squares. Multiplying this by 1,370,809,109,620,842,060,569,400 brings the total so far to 2,265,947,458,203,251,926,121,218,200.

5. Place 2 white knights on the board next.

56 choose 2 = 1,540 ways to place 2 white knights on the remaining squares. Multiplying this by 2,265,947,458,203,251,926,121,218,200 brings the total so far to 3,489,559,085,633,007,966,226,676,028,000.

6. Place 2 black knights on the board next.

54 choose 2 = 1,431 ways to place 2 black knights on the remaining squares. Multiplying this by 3,489,559,085,633,007,966,226,676,028,000 brings the total so far to 4,993,559,051,540,834,399,670,373,396,068,000.

7. Place 2 white bishops of opposite color on the board next.

This gets a little more involved. Although there are now 52 vacant squares, only 26 are available for the bishop of one color, and this leaves the other 26 available for the bishop of the other color. But what about if you promote a pawn and have two bishops on the same color? This must be accounted for in the sum as pawns are removed from the board, and it is not done here. For now, we just note that 26 squares are available to the light bishop, and 26 are available for the dark bishop, so 26 squared = 676.

26^2 = 676 ways to place 2 white bishops of opposite colors on the remaining 52 squares. Multiplying this by 4,993,559,051,540,834,399,670,373,396,068,000 brings the total so far to 3,375,645,918,841,604,054,177,172,415,741,968,000.

8. Place 2 black bishops of opposite color on the board next.

25^2 = 625 ways to place 2 black bishops of opposite colors on the remaining squares. Multiplying this by 3,375,645,918,841,604,054,177,172,415,741,968,000 brings the total so far to 2,109,778,699,276,002,533,860,732,759,838,730,000,000.

9. Place 1 white archbishop on the board next.

48 choose 1 = 48 ways to place 1 white archbishop on the remaining squares. Multiplying this by 2,109,778,699,276,002,533,860,732,759,838,730,000,000 brings the total so far to 101,269,377,565,248,121,625,315,172,472,259,040,000,000.

10. Place 1 black archbishop on the board next.

47 choose 1 = 47 ways to place 1 black archbishop on the remaining squares. Multiplying this by 101,269,377,565,248,121,625,315,172,472,259,040,000,000 brings the total so far to 4,759,660,745,566,661,716,389,813,106,196,174,880,000,000.

11. Place 1 white chancellor on the board next.

46 choose 1 = 46 ways to place 1 white chancellor on the remaining squares. Multiplying this by 4,759,660,745,566,661,716,389,813,106,196,174,880,000,000 brings the total so far to 218,944,394,296,066,438,953,931,402,885,024,044,480,000,000.

12. Place 1 black chancellor on the board next.

45 choose 1 = 45 ways to place 1 black chancellor on the remaining squares. Multiplying this by 218,944,394,296,066,438,953,931,402,885,024,044,480,000,000 brings the total so far to 9,852,497,743,322,989,752,926,913,129,826,082,001,600,000,000.

13. Place 1 white queen on the board next.

44 choose 1 = 44 ways to place 1 white queen on the remaining squares. Multiplying this by 9,852,497,743,322,989,752,926,913,129,826,082,001,600,000,000 brings the total so far to 433,509,900,706,211,549,128,784,177,712,347,608,070,400,000,000.

14. Place 1 black queen on the board next.

43 choose 1 = 43 ways to place 1 black queen on the remaining squares. Multiplying this by 433,509,900,706,211,549,128,784,177,712,347,608,070,400,000,000 brings the total so far to 18,640,925,730,367,096,612,537,719,641,630,947,147,027,200,000,000.

15. Place 1 white king on the board next.

42 choose 1 = 42 ways to place 1 white king on the remaining squares, if you ignore checks. Multiplying this by 18,640,925,730,367,096,612,537,719,641,630,947,147,027,200,000,000 brings the total so far to 782,918,880,675,418,057,726,584,224,948,499,780,175,142,400,000,000.

16. Place 1 black king on the board next.

41 choose 1 = 41 ways to place 1 black king on the remaining squares, if you ignore checks and adjacency to the other king. Multiplying this by 782,918,880,675,418,057,726,584,224,948,499,780,175,142,400,000,000 brings the total so far to 32,099,674,107,692,140,366,789,953,222,888,490,987,180,838,400,000,000.

So are we done? Not even close! This is the number of arrangements possible if no pieces were captured!! This calculation needs to be performed again for 10 pawns against 9, 9 pawns against 10, 10 pawns against 8, etc., until you get to 0 pawns for each side. Then you have to remove 1 Knight for one side, and redo the calculation again! You can see how this number would really get huge after going through all of these permutations! But then you have to iterate over all of the pawn calculations again, this time assuming that each pawn that was removed was promoted. And, recall, a pawn could promote to a queen, chancellor, archbishop, rook, bishop, or knight, so the calculations need to take each of these possibilities (2 queens for one side, 3 queens, 4 queens, 3 queens and 2 chancellors, 4 queens 3 chancellors and 2 archbishops, 6 queens + 5 rooks + 3 bishops and 3 knights, etc.!!)

32,099,674,107,692,140,366,789,953,222,888,490,987,180,838,400,000,000 is a very small number compared to this total, since we still have such a long way to go!

The first move (white moves and black replies) in regular chess can be done in 400 different combinations (20 x 20). (20 = 16 pawn moves and 4 knight moves)

If I'm not mistaken, in Gothic Chess it would be 784 (28 x 28).

I read that the total number of combinations of the first four moves in chess is 84.998.978.956 ways; while in Gothic Chess it is 1.509.030.960.338 !!!

On the other hand, In 1979, Asimov wrote Isaac Asimov's Book of Facts. On page 68, he says, "The number of possible ways of playing just the first four moves on each side in a game of chess is 318,979,564,000." This may be wrong. The number of possible ways for White to play the first move is 20 (16 pawn moves and 4 knight moves). For the first move with Black, the number is 400. For the 2nd move for white, the number of possible moves is 8,902 (5,362 distinct). For the 2nd move for Black, the number of possible moves is 197,281 (71,852 distinct). For the 3rd move for White, the number of possible moves is 4,865,617. For the 3rd move for Black, the number of possible moves is 119,060,679. For the 4th move for White, the number of possible moves is 3,195,913,043. For the 4th move for Black, the number of possible moves is 84,999,425,906. This is smaller than what Asimov says.


How about the second and the third move in gothic chess? Has anybody calculated that?

Just wanted to welcome you to the arena of Gothic Chess!

Yes, there are enormous positions in Gothic Chess and since we are going to be involved with it's infancy for a long time, this is what is going to be fun part for all of us! As we continue to talk about piece values, endgames, openings, etc. we will always be drawn to being creative. Unlike Chess where this is almost a lost phase of the game, we will continue to be energetic in finding NEW ideas in this wonderful game.

Once again welcome, and I hope to see everyone on GothicChessLive.

In the computer programming world, what you are describing is called a "perft" calculation. This is different than what I was doing above. I was calculating all of the different ways you could arrange the black and white pieces without any captures, and without illegal/impossible checks of the kings being eliminated.

Perft has been calculated to 10 plies in chess and 8 plies in Gothic Chess.

As I posted elsewhere:

Total moves (ply)

Gothic Chess


28 (1)
784 (2)
25,283 (3)
808,984 (4)
28,946,187 (5)
1,025,229,212 (6)
39,532,257,395 (7)
1,509,030,960,338 (8)

Chess

20 (1)
400 (2)
8,902 (3)
197,281 (4)
4,865,609 (5)
119,060,324 (6)
3,195,901,860 (7)
84,998,978,956 (8)
2,439,530,234,167 (9)
69,352,859,712,417 (10)

If you compare the ply 8 numbers, you can see that Gothic Chess has 17.75 times as many positions as regular chess.

Just to let you know how big that 8-ply Gothic Chess number is:

If you count the positions at a rate of 1,000,000 per second, it would take 17 days 11 hours 10 minutes 30.96 seconds to complete!