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Welcome back to Chesstalk, everyone, here is a mind-blowing post to officially put everyone back into Chess Zombie status:
Let's say you want to analyze every single unique and legal chess position in which both White and Black have 1 each of the 6 unique chess pieces: King, Queen, Rook, Bishop, Knight, Pawn. If you could analyze to completion (say using Houdini 3000) each and every such position in 1 second flat, how long would it take you?
Obviously, there is no definitive answer to this. But can we approximate it?
Yes we can. To make it clearer, let's lay it out thusly:
The numbers below each piece each represent a board square.
If we leave the pieces in the exact order they are, how many ways can we arrange the 12 boardsqures?
The answer is: 12 factorial (12!, or 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 *1). That number is 479,001,600.
Now: How many unique ways can you pick 12 board squares out of 64 unique board squares?
The number of ways of picking n distinct elements from a total of N:
P(n, N) = N!/(n! * (N-n)!)
= 64! / (12! * 52!)
However, we don't need to divide by 12! because for each pick of 12 board squares, there are 12! unique arrangements for the 12 pieces, as shown above.
Thus the number of unique arrangements of the 12 unique pieces on 12 unique board squares is
64! / (64 - 12)!
= 64! / 52!
= 53 * 54 * 55 * 56 * 57 * 58 * 59 * 60 * 61 * 62 * 63 * 64
= 1,573,144,097,507,348,889,600
Many of these positions will be illegal in standard chess for the following reasons:
(1) White and Black Kings are adjacent
(2) White or Black Pawn (or both) is on White's or Black's first rank.
(3) White and Black Kings are both in check (by any number of opposing pieces)
(4) Either White or Black King is in check from 3 or more opposing pieces
(5) Either White or Black King is in double check, and there is no possible move that could
have been made to uncover the double check (since a discovered double check is the
only double check possible in standard chess)
Of these, (5) is by far the most difficult to determine. However, (3) and (4) are also extremely
problematic because if either White's or Black's Pawn are on White's or Black's 8th rank respectively,
they are no longer Pawns but any one of the other major pieces, and could be giving check.
Can any of the items (1) to (5) be calculated mathematically?
(1) and (2), probably.
(3), (4), not without great effort possibly taking a lifetime, and
(5) almost assuredly no.
Ok... Let's assume that 73,144,097,507,348,889,600 of these arrangements constitute an illegal chess position. That's a reasonably generous allowance; it works out to 4.65% of the total positions.
I chose that number because it leaves 1,500,000,000,000,000,000,000 legal and unique chess positions with just the 6 unique White and 6 unique Black pieces.
If you could set up each legal and unique position and thoroughly analyze it in an incredible 1 second, it would take you 47,532,131,721,043 years and about 5 additional months to analyze them all.
That's 47.5 TRILLION years.
The universe is estimated to be about 13.75 billion years old. Let's call that a "Universenium".
(I looked on Wikipedia under "age of the universe" and found no named term for it. Does that mean I've actually been the first to name it?)
It would thus take you 3,457 Universeniums to analyze each and every legal and unique chess position composed of just 6 unique White and 6 unique Black pieces.
Oh... actually, double that. Each position could be White to move or Black to move.
But... there is one time-saver that might chop off some billions of years. For each position you set up and analyze, every move made that is not a capture creates another of the 1.5 sextillion unique positions. So if you make an average of say 9 such moves before a capture is made, you are actually analyzing 10 of the unique positions -- that's 9 of them you don't have to set up and analyze.
Ok, so double 3,457 Universeniums = 6,914 Universeniums (I'd like to round off and say 6,900, but 14 Universeniums is... shall we say "significant"?). Divide that by 10... about 691 Universeniums to get it all done.
But wait... some of those 9 positions we saved are bound to be duplicates so we wouldn't save quite as many positions as we thought... so let's finally do some rounding off and say a nice, round, even
700 Universeniums to analyze all the unique positions of 6 unique White pieces and 6 unique Black pieces, analyzing one such position per second.
"It was my understanding there would be no math." -- Chevy Chase
Let's say you want to analyze every single unique and legal chess position in which both White and Black have 1 each of the 6 unique chess pieces: King, Queen, Rook, Bishop, Knight, Pawn. If you could analyze to completion (say using Houdini 3000) each and every such position in 1 second flat, how long would it take you?
Obviously, there is no definitive answer to this. But can we approximate it?
Yes we can. To make it clearer, let's lay it out thusly:
Code:
WK WQ WR WB WN WP 1 2 3 4 5 6 BK BQ BR BB BN BP 7 8 9 10 11 12
If we leave the pieces in the exact order they are, how many ways can we arrange the 12 boardsqures?
The answer is: 12 factorial (12!, or 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 *1). That number is 479,001,600.
Now: How many unique ways can you pick 12 board squares out of 64 unique board squares?
The number of ways of picking n distinct elements from a total of N:
P(n, N) = N!/(n! * (N-n)!)
= 64! / (12! * 52!)
However, we don't need to divide by 12! because for each pick of 12 board squares, there are 12! unique arrangements for the 12 pieces, as shown above.
Thus the number of unique arrangements of the 12 unique pieces on 12 unique board squares is
64! / (64 - 12)!
= 64! / 52!
= 53 * 54 * 55 * 56 * 57 * 58 * 59 * 60 * 61 * 62 * 63 * 64
= 1,573,144,097,507,348,889,600
Many of these positions will be illegal in standard chess for the following reasons:
(1) White and Black Kings are adjacent
(2) White or Black Pawn (or both) is on White's or Black's first rank.
(3) White and Black Kings are both in check (by any number of opposing pieces)
(4) Either White or Black King is in check from 3 or more opposing pieces
(5) Either White or Black King is in double check, and there is no possible move that could
have been made to uncover the double check (since a discovered double check is the
only double check possible in standard chess)
Of these, (5) is by far the most difficult to determine. However, (3) and (4) are also extremely
problematic because if either White's or Black's Pawn are on White's or Black's 8th rank respectively,
they are no longer Pawns but any one of the other major pieces, and could be giving check.
Can any of the items (1) to (5) be calculated mathematically?
(1) and (2), probably.
(3), (4), not without great effort possibly taking a lifetime, and
(5) almost assuredly no.
Ok... Let's assume that 73,144,097,507,348,889,600 of these arrangements constitute an illegal chess position. That's a reasonably generous allowance; it works out to 4.65% of the total positions.
I chose that number because it leaves 1,500,000,000,000,000,000,000 legal and unique chess positions with just the 6 unique White and 6 unique Black pieces.
If you could set up each legal and unique position and thoroughly analyze it in an incredible 1 second, it would take you 47,532,131,721,043 years and about 5 additional months to analyze them all.
That's 47.5 TRILLION years.
The universe is estimated to be about 13.75 billion years old. Let's call that a "Universenium".
(I looked on Wikipedia under "age of the universe" and found no named term for it. Does that mean I've actually been the first to name it?)
It would thus take you 3,457 Universeniums to analyze each and every legal and unique chess position composed of just 6 unique White and 6 unique Black pieces.
Oh... actually, double that. Each position could be White to move or Black to move.
But... there is one time-saver that might chop off some billions of years. For each position you set up and analyze, every move made that is not a capture creates another of the 1.5 sextillion unique positions. So if you make an average of say 9 such moves before a capture is made, you are actually analyzing 10 of the unique positions -- that's 9 of them you don't have to set up and analyze.
Ok, so double 3,457 Universeniums = 6,914 Universeniums (I'd like to round off and say 6,900, but 14 Universeniums is... shall we say "significant"?). Divide that by 10... about 691 Universeniums to get it all done.
But wait... some of those 9 positions we saved are bound to be duplicates so we wouldn't save quite as many positions as we thought... so let's finally do some rounding off and say a nice, round, even
700 Universeniums to analyze all the unique positions of 6 unique White pieces and 6 unique Black pieces, analyzing one such position per second.
"It was my understanding there would be no math." -- Chevy Chase
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