Game combinations of tic-tac-toe [on hold]
How many combinations are possible in the game tic-tac-toe (Noughts and crosses)?
So for example a game which looked like: (with positions 1-9)
A1 -- B1
A2 -- B2
A3 -- --
[1][3][4][6][7] would be one combination
combinatorics combinatorial-game-theory tic-tac-toe
edited Jul 26 '16 at 22:03
Rodrigo de Azevedo
12.8k41855
asked Jan 2 '13 at 9:50
DanieldDanield
118115
put on hold as off-topic by user21820, A. Pongrácz, Gibbs, mrtaurho, Eevee Trainer Jan 7 at 2:19
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, A. Pongrácz, Gibbs, mrtaurho, Eevee Trainer
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
How many combinations are possible in the game tic-tac-toe (Noughts and crosses)?
So for example a game which looked like: (with positions 1-9)
A1 -- B1
A2 -- B2
A3 -- --
[1][3][4][6][7] would be one combination
combinatorics combinatorial-game-theory tic-tac-toe
edited Jul 26 '16 at 22:03
Rodrigo de Azevedo
12.8k41855
asked Jan 2 '13 at 9:50
DanieldDanield
118115
put on hold as off-topic by user21820, A. Pongrácz, Gibbs, mrtaurho, Eevee Trainer Jan 7 at 2:19
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, A. Pongrácz, Gibbs, mrtaurho, Eevee Trainer
If this question can be reworded to fit the rules in the help center, please edit the question.
4
se16.info/hgb/tictactoe.htm
– Daryl
Jan 2 '13 at 10:00
@Daryl, if you feel up to it, you should repackage that as an answer citing that website so this question can have an answer.
– Mark S.
Nov 22 '13 at 3:42
@MarkS. Sure thing.
– Daryl
Nov 22 '13 at 5:14
add a comment |
How many combinations are possible in the game tic-tac-toe (Noughts and crosses)?
So for example a game which looked like: (with positions 1-9)
A1 -- B1
A2 -- B2
A3 -- --
[1][3][4][6][7] would be one combination
combinatorics combinatorial-game-theory tic-tac-toe
edited Jul 26 '16 at 22:03
Rodrigo de Azevedo
12.8k41855
asked Jan 2 '13 at 9:50
DanieldDanield
118115
How many combinations are possible in the game tic-tac-toe (Noughts and crosses)?
So for example a game which looked like: (with positions 1-9)
A1 -- B1
A2 -- B2
A3 -- --
[1][3][4][6][7] would be one combination
combinatorics combinatorial-game-theory tic-tac-toe
combinatorics combinatorial-game-theory tic-tac-toe
edited Jul 26 '16 at 22:03
Rodrigo de Azevedo
12.8k41855
asked Jan 2 '13 at 9:50
DanieldDanield
118115
edited Jul 26 '16 at 22:03
Rodrigo de Azevedo
12.8k41855
asked Jan 2 '13 at 9:50
DanieldDanield
118115
edited Jul 26 '16 at 22:03
Rodrigo de Azevedo
12.8k41855
edited Jul 26 '16 at 22:03
Rodrigo de Azevedo
12.8k41855
edited Jul 26 '16 at 22:03
Rodrigo de Azevedo
12.8k41855
12.8k41855
asked Jan 2 '13 at 9:50
DanieldDanield
118115
asked Jan 2 '13 at 9:50
DanieldDanield
118115
asked Jan 2 '13 at 9:50
DanieldDanield
118115
118115
put on hold as off-topic by user21820, A. Pongrácz, Gibbs, mrtaurho, Eevee Trainer Jan 7 at 2:19
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, A. Pongrácz, Gibbs, mrtaurho, Eevee Trainer
If this question can be reworded to fit the rules in the help center, please edit the question.
put on hold as off-topic by user21820, A. Pongrácz, Gibbs, mrtaurho, Eevee Trainer Jan 7 at 2:19
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, A. Pongrácz, Gibbs, mrtaurho, Eevee Trainer
If this question can be reworded to fit the rules in the help center, please edit the question.
4
se16.info/hgb/tictactoe.htm
– Daryl
Jan 2 '13 at 10:00
@Daryl, if you feel up to it, you should repackage that as an answer citing that website so this question can have an answer.
– Mark S.
Nov 22 '13 at 3:42
@MarkS. Sure thing.
– Daryl
Nov 22 '13 at 5:14
add a comment |
4
se16.info/hgb/tictactoe.htm
– Daryl
Jan 2 '13 at 10:00
@Daryl, if you feel up to it, you should repackage that as an answer citing that website so this question can have an answer.
– Mark S.
Nov 22 '13 at 3:42
@MarkS. Sure thing.
– Daryl
Nov 22 '13 at 5:14
4
4
se16.info/hgb/tictactoe.htm
– Daryl
Jan 2 '13 at 10:00
se16.info/hgb/tictactoe.htm
– Daryl
Jan 2 '13 at 10:00
@Daryl, if you feel up to it, you should repackage that as an answer citing that website so this question can have an answer.
– Mark S.
Nov 22 '13 at 3:42
@Daryl, if you feel up to it, you should repackage that as an answer citing that website so this question can have an answer.
– Mark S.
Nov 22 '13 at 3:42
@MarkS. Sure thing.
– Daryl
Nov 22 '13 at 5:14
@MarkS. Sure thing.
– Daryl
Nov 22 '13 at 5:14
add a comment |
2 Answers
2
active
oldest
votes
This information is taken from this website.
A naive estimate would be $9!=362,880$, since there are $9$ possible first moves, $8$ for the second move, etc. This does not take into account games which finish in less than $9$ moves.
- Ending on the $5^text{th}$ move: $1,440$ possibilities
- Ending on the $6^text{th}$ move: $5,328$ possibilities
- Ending on the $7^text{th}$ move: $47,952$ possibilities
- Ending on the $8^text{th}$ move: $72,576$ possibilities
- Ending on the $9^text{th}$ move: $127,872$ possibilities
This gives a total of $255168$ possible games. This calculation doesn't take into account symmetry in the game.
answered Nov 22 '13 at 5:21
DarylDaryl
4,74231835
Accounting for symmetry, this can quickly be reduced by a factor of 6, as there are only 12 possible two move openings, not 8*9=72.
– Pieter Geerkens
Nov 22 '13 at 5:32
As that linked page makes clear at the bottom, symmetry allows a reduction by a factor of $8$ (i.e. rotations and reflections of a square) to $31896$, but arguably by a greater amount to $26830$.
– Henry
Jan 2 '15 at 11:18
I am assuming I am asking a naive question here, but why isn't the answer $9! - sumlimits_{i=5}^{8} m_i = 235584$, where, $m_i$ are the number of games ending on $i$ moves (the numbers above). (From the overestimation of the games that take all 9 moves we subtract the ones that require less)
– kyriakosSt
Dec 3 '17 at 23:47
add a comment |
I will say that the board combinations are 3^9, which is 19683 possibilities, and 2032 winning positions. The answer of 9! is related to how many ways we have to fell all the positions, rather than the possible combinations.
I have answered this question already in another post, please see the next link: https://stackoverflow.com/a/54035004/5117217
Cheers!
answered Jan 4 at 20:07
user5117217user5117217
1
New contributor
user5117217 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
This information is taken from this website.
A naive estimate would be $9!=362,880$, since there are $9$ possible first moves, $8$ for the second move, etc. This does not take into account games which finish in less than $9$ moves.
- Ending on the $5^text{th}$ move: $1,440$ possibilities
- Ending on the $6^text{th}$ move: $5,328$ possibilities
- Ending on the $7^text{th}$ move: $47,952$ possibilities
- Ending on the $8^text{th}$ move: $72,576$ possibilities
- Ending on the $9^text{th}$ move: $127,872$ possibilities
This gives a total of $255168$ possible games. This calculation doesn't take into account symmetry in the game.
answered Nov 22 '13 at 5:21
DarylDaryl
4,74231835
Accounting for symmetry, this can quickly be reduced by a factor of 6, as there are only 12 possible two move openings, not 8*9=72.
– Pieter Geerkens
Nov 22 '13 at 5:32
As that linked page makes clear at the bottom, symmetry allows a reduction by a factor of $8$ (i.e. rotations and reflections of a square) to $31896$, but arguably by a greater amount to $26830$.
– Henry
Jan 2 '15 at 11:18
I am assuming I am asking a naive question here, but why isn't the answer $9! - sumlimits_{i=5}^{8} m_i = 235584$, where, $m_i$ are the number of games ending on $i$ moves (the numbers above). (From the overestimation of the games that take all 9 moves we subtract the ones that require less)
– kyriakosSt
Dec 3 '17 at 23:47
add a comment |
This information is taken from this website.
A naive estimate would be $9!=362,880$, since there are $9$ possible first moves, $8$ for the second move, etc. This does not take into account games which finish in less than $9$ moves.
- Ending on the $5^text{th}$ move: $1,440$ possibilities
- Ending on the $6^text{th}$ move: $5,328$ possibilities
- Ending on the $7^text{th}$ move: $47,952$ possibilities
- Ending on the $8^text{th}$ move: $72,576$ possibilities
- Ending on the $9^text{th}$ move: $127,872$ possibilities
This gives a total of $255168$ possible games. This calculation doesn't take into account symmetry in the game.
answered Nov 22 '13 at 5:21
DarylDaryl
4,74231835
Accounting for symmetry, this can quickly be reduced by a factor of 6, as there are only 12 possible two move openings, not 8*9=72.
– Pieter Geerkens
Nov 22 '13 at 5:32
As that linked page makes clear at the bottom, symmetry allows a reduction by a factor of $8$ (i.e. rotations and reflections of a square) to $31896$, but arguably by a greater amount to $26830$.
– Henry
Jan 2 '15 at 11:18
I am assuming I am asking a naive question here, but why isn't the answer $9! - sumlimits_{i=5}^{8} m_i = 235584$, where, $m_i$ are the number of games ending on $i$ moves (the numbers above). (From the overestimation of the games that take all 9 moves we subtract the ones that require less)
– kyriakosSt
Dec 3 '17 at 23:47
add a comment |
This information is taken from this website.
A naive estimate would be $9!=362,880$, since there are $9$ possible first moves, $8$ for the second move, etc. This does not take into account games which finish in less than $9$ moves.
- Ending on the $5^text{th}$ move: $1,440$ possibilities
- Ending on the $6^text{th}$ move: $5,328$ possibilities
- Ending on the $7^text{th}$ move: $47,952$ possibilities
- Ending on the $8^text{th}$ move: $72,576$ possibilities
- Ending on the $9^text{th}$ move: $127,872$ possibilities
This gives a total of $255168$ possible games. This calculation doesn't take into account symmetry in the game.
answered Nov 22 '13 at 5:21
DarylDaryl
4,74231835
This information is taken from this website.
A naive estimate would be $9!=362,880$, since there are $9$ possible first moves, $8$ for the second move, etc. This does not take into account games which finish in less than $9$ moves.
- Ending on the $5^text{th}$ move: $1,440$ possibilities
- Ending on the $6^text{th}$ move: $5,328$ possibilities
- Ending on the $7^text{th}$ move: $47,952$ possibilities
- Ending on the $8^text{th}$ move: $72,576$ possibilities
- Ending on the $9^text{th}$ move: $127,872$ possibilities
This gives a total of $255168$ possible games. This calculation doesn't take into account symmetry in the game.
answered Nov 22 '13 at 5:21
DarylDaryl
4,74231835
answered Nov 22 '13 at 5:21
DarylDaryl
4,74231835
answered Nov 22 '13 at 5:21
DarylDaryl
4,74231835
answered Nov 22 '13 at 5:21
DarylDaryl
4,74231835
4,74231835
Accounting for symmetry, this can quickly be reduced by a factor of 6, as there are only 12 possible two move openings, not 8*9=72.
– Pieter Geerkens
Nov 22 '13 at 5:32
As that linked page makes clear at the bottom, symmetry allows a reduction by a factor of $8$ (i.e. rotations and reflections of a square) to $31896$, but arguably by a greater amount to $26830$.
– Henry
Jan 2 '15 at 11:18
I am assuming I am asking a naive question here, but why isn't the answer $9! - sumlimits_{i=5}^{8} m_i = 235584$, where, $m_i$ are the number of games ending on $i$ moves (the numbers above). (From the overestimation of the games that take all 9 moves we subtract the ones that require less)
– kyriakosSt
Dec 3 '17 at 23:47
add a comment |
Accounting for symmetry, this can quickly be reduced by a factor of 6, as there are only 12 possible two move openings, not 8*9=72.
– Pieter Geerkens
Nov 22 '13 at 5:32
As that linked page makes clear at the bottom, symmetry allows a reduction by a factor of $8$ (i.e. rotations and reflections of a square) to $31896$, but arguably by a greater amount to $26830$.
– Henry
Jan 2 '15 at 11:18
I am assuming I am asking a naive question here, but why isn't the answer $9! - sumlimits_{i=5}^{8} m_i = 235584$, where, $m_i$ are the number of games ending on $i$ moves (the numbers above). (From the overestimation of the games that take all 9 moves we subtract the ones that require less)
– kyriakosSt
Dec 3 '17 at 23:47
Accounting for symmetry, this can quickly be reduced by a factor of 6, as there are only 12 possible two move openings, not 8*9=72.
– Pieter Geerkens
Nov 22 '13 at 5:32
Accounting for symmetry, this can quickly be reduced by a factor of 6, as there are only 12 possible two move openings, not 8*9=72.
– Pieter Geerkens
Nov 22 '13 at 5:32
As that linked page makes clear at the bottom, symmetry allows a reduction by a factor of $8$ (i.e. rotations and reflections of a square) to $31896$, but arguably by a greater amount to $26830$.
– Henry
Jan 2 '15 at 11:18
As that linked page makes clear at the bottom, symmetry allows a reduction by a factor of $8$ (i.e. rotations and reflections of a square) to $31896$, but arguably by a greater amount to $26830$.
– Henry
Jan 2 '15 at 11:18
I am assuming I am asking a naive question here, but why isn't the answer $9! - sumlimits_{i=5}^{8} m_i = 235584$, where, $m_i$ are the number of games ending on $i$ moves (the numbers above). (From the overestimation of the games that take all 9 moves we subtract the ones that require less)
– kyriakosSt
Dec 3 '17 at 23:47
I am assuming I am asking a naive question here, but why isn't the answer $9! - sumlimits_{i=5}^{8} m_i = 235584$, where, $m_i$ are the number of games ending on $i$ moves (the numbers above). (From the overestimation of the games that take all 9 moves we subtract the ones that require less)
– kyriakosSt
Dec 3 '17 at 23:47
add a comment |
I will say that the board combinations are 3^9, which is 19683 possibilities, and 2032 winning positions. The answer of 9! is related to how many ways we have to fell all the positions, rather than the possible combinations.
I have answered this question already in another post, please see the next link: https://stackoverflow.com/a/54035004/5117217
Cheers!
answered Jan 4 at 20:07
user5117217user5117217
1
New contributor
user5117217 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
I will say that the board combinations are 3^9, which is 19683 possibilities, and 2032 winning positions. The answer of 9! is related to how many ways we have to fell all the positions, rather than the possible combinations.
I have answered this question already in another post, please see the next link: https://stackoverflow.com/a/54035004/5117217
Cheers!
answered Jan 4 at 20:07
user5117217user5117217
1
New contributor
user5117217 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
I will say that the board combinations are 3^9, which is 19683 possibilities, and 2032 winning positions. The answer of 9! is related to how many ways we have to fell all the positions, rather than the possible combinations.
I have answered this question already in another post, please see the next link: https://stackoverflow.com/a/54035004/5117217
Cheers!
answered Jan 4 at 20:07
user5117217user5117217
1
New contributor
user5117217 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I will say that the board combinations are 3^9, which is 19683 possibilities, and 2032 winning positions. The answer of 9! is related to how many ways we have to fell all the positions, rather than the possible combinations.
I have answered this question already in another post, please see the next link: https://stackoverflow.com/a/54035004/5117217
Cheers!
answered Jan 4 at 20:07
user5117217user5117217
1
New contributor
user5117217 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Jan 4 at 20:07
user5117217user5117217
1
New contributor
user5117217 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Jan 4 at 20:07
user5117217user5117217
1
answered Jan 4 at 20:07
user5117217user5117217
1
1
New contributor
user5117217 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
user5117217 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
user5117217 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
4
se16.info/hgb/tictactoe.htm
– Daryl
Jan 2 '13 at 10:00
@Daryl, if you feel up to it, you should repackage that as an answer citing that website so this question can have an answer.
– Mark S.
Nov 22 '13 at 3:42
@MarkS. Sure thing.
– Daryl
Nov 22 '13 at 5:14