Puzzle: mathematical puzzle
Puzzle: mathematical puzzle
Given that each player at start of a game has 20 possible opening moves, the number of possible board positions after 1 move each is 400.
What is the MAXIMUM number of legal board positions after 2 moves each ?
What is the MAXIMUM number of legal board positions after 2 moves each ?
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- Uranium
- Posts: 266
- Joined: Mon Apr 12, 1999 3:52 am
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Chess positions
Hi,
I just did a Google search for ("Chess positions" 400).
Howstuffworks has a site of "How Chess Works" and it says probably about 160,000 after 2 moves (20x20x~20x~20).
Another site at MathWorld or Works listed the positions as: 20, 400, 5362, 71852, ...
I'm fiddling with some of it at work. Unfortunately some of white's second moves are obviously dependent on Black's first move but the rest isn't too difficult to write out (for instance, if white plays a3 1st move then regardless of what black does, white will only have 19 possibilities for a 2nd move - playing a4 adds the Ra3 move, Na3 move and a possible axb5 move for a total of 22 possibilities).
Here's what I've got so far....
a3->19
a4->22
b3->21
b4->23
c3->21
c4->24
d3->27
d4->30
e3->31
e4->33
f3->18
f4->19
Gotta get back to work now.
I just did a Google search for ("Chess positions" 400).
Howstuffworks has a site of "How Chess Works" and it says probably about 160,000 after 2 moves (20x20x~20x~20).
Another site at MathWorld or Works listed the positions as: 20, 400, 5362, 71852, ...
I'm fiddling with some of it at work. Unfortunately some of white's second moves are obviously dependent on Black's first move but the rest isn't too difficult to write out (for instance, if white plays a3 1st move then regardless of what black does, white will only have 19 possibilities for a 2nd move - playing a4 adds the Ra3 move, Na3 move and a possible axb5 move for a total of 22 possibilities).
Here's what I've got so far....
a3->19
a4->22
b3->21
b4->23
c3->21
c4->24
d3->27
d4->30
e3->31
e4->33
f3->18
f4->19
Gotta get back to work now.
Maybe you miss the point
The question was not "how many combinations of moves" but how many differennt positions there could be - imagine taking a photograph after move 2. How many different photos could you have. I think the answer is less than 900
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- Uranium
- Posts: 266
- Joined: Mon Apr 12, 1999 3:52 am
- Location: NW England
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Re: Maybe you miss the point
must be more than 900 postitions white can make 16 different pawn movs for hsis first move and black can make 16 reposnses back for his first move which is 256 unique positions after one move each already not counting the several knight moves that can be madenerob wrote:The question was not "how many combinations of moves" but how many differennt positions there could be - imagine taking a photograph after move 2. How many different photos could you have. I think the answer is less than 900
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I did something similar to what was done earlier - for each first move, how many possible 2nd moves. Then I added them up and got 440. There would be 440 possible moves, if the opponents move did not give additional moves or restrict any moves.
The next step was to remove the moves where the position is identical. If my thinking is right, that is done by determining the number of original moves that are still possible after the first move.
If a knight is moved first there are 16 possible original moves still available - the original 20 minus the two knight moves, plus the two moves of the pawn the knight is now blocking.
For most pawn moves there are 18 possible original moves. (20 minus the pawns two original moves). For four of the pawn moves it is blocking a knight move, leaving 17 moves.
Doing the math... 4 (knight moves) * 16 + 4 (pawn moves that block a knight) * 17 + 12 (other pawn moves) * 18 = 348. This is the number of moves where there is another combination that ends up in the same position. Divide that by 2 to get 174. Subtract that from the 440 and you get 266. That is the number of possible positions (assuming all of my math is right) that is possible after 2 moves, if the opponent does not give additional moves or restrict any moves. Square that and it is 70,756 possible possitions for both sides.
The number 71,852 has been mentioned as the correct number of positions. My calculations did not take into account the effects of opponents moves. If both numbers are correct then there is a little over 1,000 additional positions possible because of opponents moves.
The next step was to remove the moves where the position is identical. If my thinking is right, that is done by determining the number of original moves that are still possible after the first move.
If a knight is moved first there are 16 possible original moves still available - the original 20 minus the two knight moves, plus the two moves of the pawn the knight is now blocking.
For most pawn moves there are 18 possible original moves. (20 minus the pawns two original moves). For four of the pawn moves it is blocking a knight move, leaving 17 moves.
Doing the math... 4 (knight moves) * 16 + 4 (pawn moves that block a knight) * 17 + 12 (other pawn moves) * 18 = 348. This is the number of moves where there is another combination that ends up in the same position. Divide that by 2 to get 174. Subtract that from the 440 and you get 266. That is the number of possible positions (assuming all of my math is right) that is possible after 2 moves, if the opponent does not give additional moves or restrict any moves. Square that and it is 70,756 possible possitions for both sides.
The number 71,852 has been mentioned as the correct number of positions. My calculations did not take into account the effects of opponents moves. If both numbers are correct then there is a little over 1,000 additional positions possible because of opponents moves.