Combinations and permutations - more math than chess

[John Coleman]

Two chess players can be paired only 1 way: (1-2)

Four players can be paired 3 ways: (1-2 and 3-4) (1-3 and 2-4) (1-4 and 2-3)

Six players can be paired 15 ways: I'll spare my typing finger, and not list them; trust me.

Eight players? How many ways can eight players be paired? Does anyone know the formula for the possible combinations for n players, where n is even? How about where n is uneven?

[Peter McKillop]

Re: Combinations and permutations - more math than chess

Two chess players can be paired only 1 way: (1-2)

Four players can be paired 3 ways: (1-2 and 3-4) (1-3 and 2-4) (1-4 and 2-3)

Six players can be paired 15 ways: I'll spare my typing finger, and not list them; trust me.

Eight players? How many ways can eight players be paired? Does anyone know the formula for the possible combinations for n players, where n is even? How about where n is uneven?

Hi John. Since colour is an important part of the pairing process (2-1 and 1-2), why aren't you interested in in permutations? For combinations, I think the number you're looking for is 28 (but I forget the formula). :)

https://en.wikipedia.org/wiki/Combination

[Tom O'Donnell]

Re: Combinations and permutations - more math than chess

Two chess players can be paired only 1 way: (1-2)

Four players can be paired 3 ways: (1-2 and 3-4) (1-3 and 2-4) (1-4 and 2-3)

Six players can be paired 15 ways: I'll spare my typing finger, and not list them; trust me.

Eight players? How many ways can eight players be paired? Does anyone know the formula for the possible combinations for n players, where n is even? How about where n is uneven?

The 15 setups for 6 players:

12 34 56
12 35 46
12 36 45
13 24 56
13 25 46
13 26 45
14 23 56
14 25 36
14 26 35
15 23 46
15 24 36
15 26 34
16 23 45
16 24 35
16 25 34

I think the answer for 8 players is 105.

My reasoning:

1 - 2 players
1x3 - 4 players
1x3x5 - 6 players
1x3x5x7 - 8 players


All 1 v 2 pairings, I think:

12 34 56 78
12 34 57 68
12 34 58 67
12 35 46 78
12 35 47 68
12 35 48 67
12 36 45 78
12 36 47 68
12 36 48 57
12 37 45 68
12 37 46 58
12 37 48 56
12 38 45 67
12 38 46 57
12 38 47 56 = 15 different setups.

So you would have the same sorts of pairings with 13, 14, etc. all the way to 18 (seven different groups); 7 x 15 = 105.

[Tom O'Donnell]

Re: Combinations and permutations - more math than chess

I might add that I don't see any difference between say 7 or 8 players. Wouldn't it be the same number of possible pairings (i.e. 105)?

[Peter McKillop]

Re: Combinations and permutations - more math than chess

The 15 setups for 6 players:

12 34 56
12 35 46
12 36 45
13 24 56
13 25 46
13 26 45
14 23 56
14 25 36
14 26 35
15 23 46
15 24 36
15 26 34
16 23 45
16 24 35
16 25 34

I think the answer for 8 players is 105.

My reasoning:

1 - 2 players
1x3 - 4 players
1x3x5 - 6 players
1x3x5x7 - 8 players


All 1 v 2 pairings, I think:

12 34 56 78
12 34 57 68
12 34 58 67
12 35 46 78
12 35 47 68
12 35 48 67
12 36 45 78
12 36 47 68
12 36 48 57
12 37 45 68
12 37 46 58
12 37 48 56
12 38 45 67
12 38 46 57
12 38 47 56 = 15 different setups.

So you would have the same sorts of pairings with 13, 14, etc. all the way to 18 (seven different groups); 7 x 15 = 105.

Since I had forgotten the formula for the number of combinations of eight things combined two at a time (it's been half a century since grade 11 math), here's how I came up with 28:

Player 1 can be paired with each of 7 other players.
Player 2 can be paired with each of 6 other players (the pairing of 1 and 2 was counted above).
Player 3 .... 5 other players.
.... and so on.

i.e. 7 + 6 +5 +4 +3 + 2 + 1 = 28

[Peter McKillop]

Re: Combinations and permutations - more math than chess

John, n would always be even because you have in reserve, for uneven situations, the player called 'bye'.

[Ken MacDonald]

Re: Combinations and permutations - more math than chess

Since I had forgotten the formula for the number of combinations of eight things combined two at a time (it's been half a century since grade 11 math), here's how I came up with 28:

Player 1 can be paired with each of 7 other players.
Player 2 can be paired with each of 6 other players (the pairing of 1 and 2 was counted above).
Player 3 .... 5 other players.
.... and so on.

i.e. 7 + 6 +5 +4 +3 + 2 + 1 = 28

No mathematician am I, but, I always thought the formula would be (# of players) x (# of players less 1) divided by 2.

So, 8 players would be 8 x 7 \ 2 = 28

A 15 player event would be 15 x 14 \ 2 = 105.

This is assuming you meant a round robin of x number of players playing each participant once to obtain the number of games that would be played.

[John Coleman]

Re: Combinations and permutations - more math than chess

I might add that I don't see any difference between say 7 or 8 players. Wouldn't it be the same number of possible pairings (i.e. 105)?

Thanks Tom, I came up with ths same number, but I couldn't prove it.

Before I retired from chess organising, I developed a pairing system for kids which does not require that all round start at the same time. In fact, we don't even have rounds. We get 1400+ kids at our big event each year. We don't assign colours, the kids pick colours (chosing a pawn) themselves. I'm writing a pairing program to computerise this system, by way of keeping my brain active. It's an adventure.

[Peter McKillop]

Re: Combinations and permutations - more math than chess

No mathematician am I, but, I always thought the formula would be (# of players) x (# of players less 1) divided by 2.

So, 8 players would be 8 x 7 \ 2 = 28

A 15 player event would be 15 x 14 \ 2 = 105.

This is assuming you meant a round robin of x number of players playing each participant once to obtain the number of games that would be played.

Yes, you're right about the formula for n things combined 2 at a time.

[Andrew Peredun]

Re: Combinations and permutations - more math than chess

n is any even number of players (just add a bye for odd numbers to make them all even). Then P is the number of possible pairing arrangements for any individual round (assuming your players then randomize for white & black).

P(n) = 1 * 3 * 5 * ... * (n-1)

This gives us:

P(2) = 1
P(4) = 1*3 = 3
P(6) = 1*3*5 = 15
P(8) = 1*3*5*7 = 105
P(10) = 1*3*5*7*9 = 945
etc.
Tom O'Donnell's post does a good job of explaining the reasoning.

This is different than the number of ways to select a group of two people out of a group of n, which as mentioned already is n(n-1)/2, and gives you the total number of games that would be played in a single round robin tournament.

[Peter McKillop]

Re: Combinations and permutations - more math than chess

n is any even number of players (just add a bye for odd numbers to make them all even). Then P is the number of possible pairing arrangements for any individual round (assuming your players then randomize for white & black).

P(n) = 1 * 3 * 5 * ... * (n-1)

This gives us:

P(2) = 1
P(4) = 1*3 = 3
P(6) = 1*3*5 = 15
P(8) = 1*3*5*7 = 105
P(10) = 1*3*5*7*9 = 945
etc.
Tom O'Donnell's post does a good job of explaining the reasoning.

This is different than the number of ways to select a group of two people out of a group of n, which as mentioned already is n(n-1)/2, and gives you the total number of games that would be played in a single round robin tournament.

Thanks, Andrew. I misunderstood what John was looking for.

[Aris Marghetis]

Re: Combinations and permutations - more math than chess

...

Before I retired from chess organising, I developed a pairing system for kids which does not require that all round start at the same time. In fact, we don't even have rounds. We get 1400+ kids at our big event each year. We don't assign colours, the kids pick colours (chosing a pawn) themselves. I'm writing a pairing program to computerise this system, by way of keeping my brain active. It's an adventure.

Hi John, I would be interested in such a program. Thanks and regards, Aris.