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Game schedule where each player meets only once

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.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;

2

2

$begingroup$

I am trying to generate a game schedule where $k$ players at the time, out of $n$ participants, meet in each game, but any player meets another player only once. In each game, with $k$ simultaneous players, the $k$ players compete against each other, there are no multi-player teams.

A 'raw and basic' method for a $(7,3)$-game is shown below, but there must be a more elegant way in Mathematica. It is not very customisable for a general $(n,k)$-game. TIA.

n = 7;
mat = Table[1, x, n, y, n];
For[i = 1, i <= n, i++,
 For[j = i + 1, j <= n, j++,
 For[k = j + 1, k <= n, k++,
 If[mat[[i,j]]*mat[[j,i]]*mat[[i,k]]*mat[[k,i]]*mat[[j,k]]*mat[[k,j]] == 1,
 p = i, j, k;
 Print [p];
 mat[[i,j]] = 0;
 mat[[j,i]] = 0;
 mat[[i,k]] = 0;
 mat[[k,i]] = 0;
 mat[[j,k]] = 0;
 mat[[k,j]] = 0;
 ]
 ]
 ]
]

Result: 1,2,3, 1,4,5, 1,6,7, 2,4,6, 2,5,7, 3,4,7, 3,5,6.

syntax

share|improve this question

asked 8 hours ago

mf67mf67

17566 bronze badges

$endgroup$

add a comment | 

2

2

$begingroup$

I am trying to generate a game schedule where $k$ players at the time, out of $n$ participants, meet in each game, but any player meets another player only once. In each game, with $k$ simultaneous players, the $k$ players compete against each other, there are no multi-player teams.

A 'raw and basic' method for a $(7,3)$-game is shown below, but there must be a more elegant way in Mathematica. It is not very customisable for a general $(n,k)$-game. TIA.

n = 7;
mat = Table[1, x, n, y, n];
For[i = 1, i <= n, i++,
 For[j = i + 1, j <= n, j++,
 For[k = j + 1, k <= n, k++,
 If[mat[[i,j]]*mat[[j,i]]*mat[[i,k]]*mat[[k,i]]*mat[[j,k]]*mat[[k,j]] == 1,
 p = i, j, k;
 Print [p];
 mat[[i,j]] = 0;
 mat[[j,i]] = 0;
 mat[[i,k]] = 0;
 mat[[k,i]] = 0;
 mat[[j,k]] = 0;
 mat[[k,j]] = 0;
 ]
 ]
 ]
]

Result: 1,2,3, 1,4,5, 1,6,7, 2,4,6, 2,5,7, 3,4,7, 3,5,6.

syntax

share|improve this question

asked 8 hours ago

mf67mf67

17566 bronze badges

$endgroup$

add a comment | 

$begingroup$

I am trying to generate a game schedule where $k$ players at the time, out of $n$ participants, meet in each game, but any player meets another player only once. In each game, with $k$ simultaneous players, the $k$ players compete against each other, there are no multi-player teams.

A 'raw and basic' method for a $(7,3)$-game is shown below, but there must be a more elegant way in Mathematica. It is not very customisable for a general $(n,k)$-game. TIA.

n = 7;
mat = Table[1, x, n, y, n];
For[i = 1, i <= n, i++,
 For[j = i + 1, j <= n, j++,
 For[k = j + 1, k <= n, k++,
 If[mat[[i,j]]*mat[[j,i]]*mat[[i,k]]*mat[[k,i]]*mat[[j,k]]*mat[[k,j]] == 1,
 p = i, j, k;
 Print [p];
 mat[[i,j]] = 0;
 mat[[j,i]] = 0;
 mat[[i,k]] = 0;
 mat[[k,i]] = 0;
 mat[[j,k]] = 0;
 mat[[k,j]] = 0;
 ]
 ]
 ]
]

Result: 1,2,3, 1,4,5, 1,6,7, 2,4,6, 2,5,7, 3,4,7, 3,5,6.

syntax

share|improve this question

asked 8 hours ago

mf67mf67

17566 bronze badges

$endgroup$

n = 7;
mat = Table[1, x, n, y, n];
For[i = 1, i <= n, i++,
 For[j = i + 1, j <= n, j++,
 For[k = j + 1, k <= n, k++,
 If[mat[[i,j]]*mat[[j,i]]*mat[[i,k]]*mat[[k,i]]*mat[[j,k]]*mat[[k,j]] == 1,
 p = i, j, k;
 Print [p];
 mat[[i,j]] = 0;
 mat[[j,i]] = 0;
 mat[[i,k]] = 0;
 mat[[k,i]] = 0;
 mat[[j,k]] = 0;
 mat[[k,j]] = 0;
 ]
 ]
 ]
]

share|improve this question

asked 8 hours ago

mf67mf67

17566 bronze badges

share|improve this question

asked 8 hours ago

mf67mf67

17566 bronze badges

asked 8 hours ago

mf67mf67

17566 bronze badges

asked 8 hours ago

mf67mf67

17566 bronze badges

2 Answers
2

active

oldest

votes

6

$begingroup$

n = 7;
k = 3;

DeleteDuplicates[Subsets[Range @ n, k], Length[Intersection[##]] >= 2 &]

1, 2, 3, 1, 4, 5, 1, 6, 7, 2, 4, 6, 2, 5, 7, 3, 4, 7, 3, 5, 6

share|improve this answer

answered 6 hours ago

kglrkglr

209k1010 gold badges241241 silver badges478478 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Very compact and 'neat'. Could you please explain what it does? I tried to 'decompile' it but had no luck understanding some of the 'components'.
  $endgroup$
  – mf67
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @mf67, Subsets[Range @ n, k] gives all k-player games, a list of k-tuples , g1,g2,.... Scanning this list from left to right DeleteDuplicates step eliminates all games $g_k$ that contains two or more players from $g_j$ (for $j<k$).
  $endgroup$
  – kglr
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I see. Much clearer now. The examples on the Mathematica help pages are often (too) short and do not display the more 'intricate' commands given on this site.
  $endgroup$
  – mf67
  5 hours ago
  

  
  
  
  

add a comment | 

1

$begingroup$

n = 7;
k = 3;

list all possible games:

g = Subsets[Range[n], k];

Find a maximal-size clique of games that don't overlap:

First@FindClique[AdjacencyGraph[Outer[Boole[Length[Intersection[##]] <= 1] &, g, g, 1]]]
(* 1, 10, 15, 21, 24, 28, 29 *)

Which games are these?

g[[%]]
(* 1, 2, 3, 1, 4, 5, 1, 6, 7, 2, 4, 6, 2, 5, 7, 3, 4, 7, 3, 5, 6 *)

This method gives the same result as @kglr's but is much slower, so I don't recommend using it. You can view it as a proof of the other method.

share|improve this answer

edited 6 hours ago

answered 6 hours ago

RomanRoman

15.2k11 gold badge2121 silver badges5252 bronze badges

$endgroup$

add a comment | 

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6

$begingroup$

n = 7;
k = 3;

DeleteDuplicates[Subsets[Range @ n, k], Length[Intersection[##]] >= 2 &]

1, 2, 3, 1, 4, 5, 1, 6, 7, 2, 4, 6, 2, 5, 7, 3, 4, 7, 3, 5, 6

share|improve this answer

answered 6 hours ago

kglrkglr

209k1010 gold badges241241 silver badges478478 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Very compact and 'neat'. Could you please explain what it does? I tried to 'decompile' it but had no luck understanding some of the 'components'.
  $endgroup$
  – mf67
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @mf67, Subsets[Range @ n, k] gives all k-player games, a list of k-tuples , g1,g2,.... Scanning this list from left to right DeleteDuplicates step eliminates all games $g_k$ that contains two or more players from $g_j$ (for $j<k$).
  $endgroup$
  – kglr
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I see. Much clearer now. The examples on the Mathematica help pages are often (too) short and do not display the more 'intricate' commands given on this site.
  $endgroup$
  – mf67
  5 hours ago
  

  
  
  
  

add a comment | 

6

$begingroup$

n = 7;
k = 3;

DeleteDuplicates[Subsets[Range @ n, k], Length[Intersection[##]] >= 2 &]

1, 2, 3, 1, 4, 5, 1, 6, 7, 2, 4, 6, 2, 5, 7, 3, 4, 7, 3, 5, 6

share|improve this answer

answered 6 hours ago

kglrkglr

209k1010 gold badges241241 silver badges478478 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Very compact and 'neat'. Could you please explain what it does? I tried to 'decompile' it but had no luck understanding some of the 'components'.
  $endgroup$
  – mf67
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @mf67, Subsets[Range @ n, k] gives all k-player games, a list of k-tuples , g1,g2,.... Scanning this list from left to right DeleteDuplicates step eliminates all games $g_k$ that contains two or more players from $g_j$ (for $j<k$).
  $endgroup$
  – kglr
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I see. Much clearer now. The examples on the Mathematica help pages are often (too) short and do not display the more 'intricate' commands given on this site.
  $endgroup$
  – mf67
  5 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

n = 7;
k = 3;

DeleteDuplicates[Subsets[Range @ n, k], Length[Intersection[##]] >= 2 &]

1, 2, 3, 1, 4, 5, 1, 6, 7, 2, 4, 6, 2, 5, 7, 3, 4, 7, 3, 5, 6

share|improve this answer

answered 6 hours ago

kglrkglr

209k1010 gold badges241241 silver badges478478 bronze badges

$endgroup$

n = 7;
k = 3;

DeleteDuplicates[Subsets[Range @ n, k], Length[Intersection[##]] >= 2 &]

share|improve this answer

answered 6 hours ago

kglrkglr

209k1010 gold badges241241 silver badges478478 bronze badges

answered 6 hours ago

kglrkglr

209k1010 gold badges241241 silver badges478478 bronze badges

answered 6 hours ago

kglrkglr

209k1010 gold badges241241 silver badges478478 bronze badges

1

$begingroup$

n = 7;
k = 3;

list all possible games:

g = Subsets[Range[n], k];

Find a maximal-size clique of games that don't overlap:

First@FindClique[AdjacencyGraph[Outer[Boole[Length[Intersection[##]] <= 1] &, g, g, 1]]]
(* 1, 10, 15, 21, 24, 28, 29 *)

Which games are these?

g[[%]]
(* 1, 2, 3, 1, 4, 5, 1, 6, 7, 2, 4, 6, 2, 5, 7, 3, 4, 7, 3, 5, 6 *)

This method gives the same result as @kglr's but is much slower, so I don't recommend using it. You can view it as a proof of the other method.

share|improve this answer

edited 6 hours ago

answered 6 hours ago

RomanRoman

15.2k11 gold badge2121 silver badges5252 bronze badges

$endgroup$

add a comment | 

1

$begingroup$

n = 7;
k = 3;

list all possible games:

g = Subsets[Range[n], k];

Find a maximal-size clique of games that don't overlap:

First@FindClique[AdjacencyGraph[Outer[Boole[Length[Intersection[##]] <= 1] &, g, g, 1]]]
(* 1, 10, 15, 21, 24, 28, 29 *)

Which games are these?

g[[%]]
(* 1, 2, 3, 1, 4, 5, 1, 6, 7, 2, 4, 6, 2, 5, 7, 3, 4, 7, 3, 5, 6 *)

This method gives the same result as @kglr's but is much slower, so I don't recommend using it. You can view it as a proof of the other method.

share|improve this answer

edited 6 hours ago

answered 6 hours ago

RomanRoman

15.2k11 gold badge2121 silver badges5252 bronze badges

$endgroup$

add a comment | 

$begingroup$

n = 7;
k = 3;

list all possible games:

g = Subsets[Range[n], k];

Find a maximal-size clique of games that don't overlap:

First@FindClique[AdjacencyGraph[Outer[Boole[Length[Intersection[##]] <= 1] &, g, g, 1]]]
(* 1, 10, 15, 21, 24, 28, 29 *)

Which games are these?

g[[%]]
(* 1, 2, 3, 1, 4, 5, 1, 6, 7, 2, 4, 6, 2, 5, 7, 3, 4, 7, 3, 5, 6 *)

This method gives the same result as @kglr's but is much slower, so I don't recommend using it. You can view it as a proof of the other method.

share|improve this answer

edited 6 hours ago

answered 6 hours ago

RomanRoman

15.2k11 gold badge2121 silver badges5252 bronze badges

$endgroup$

n = 7;
k = 3;

g = Subsets[Range[n], k];

First@FindClique[AdjacencyGraph[Outer[Boole[Length[Intersection[##]] <= 1] &, g, g, 1]]]
(* 1, 10, 15, 21, 24, 28, 29 *)

g[[%]]
(* 1, 2, 3, 1, 4, 5, 1, 6, 7, 2, 4, 6, 2, 5, 7, 3, 4, 7, 3, 5, 6 *)

share|improve this answer

edited 6 hours ago

answered 6 hours ago

RomanRoman

15.2k11 gold badge2121 silver badges5252 bronze badges

answered 6 hours ago

RomanRoman

15.2k11 gold badge2121 silver badges5252 bronze badges

answered 6 hours ago

RomanRoman

15.2k11 gold badge2121 silver badges5252 bronze badges