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Basic Math/Another twist to the golf pairing question

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Expert: Josh - 5/9/2011


Question
Came across your answer after a lot of stumbling around on the web.  We have a total of 12 players and 5 rounds.  So challenge is to make foursomes as unique as possible.  Additional twist is that there will be 26 pairs playing head-to-head matches over this period so they have to be together.  For example:

R1: AB, CD, EF, GH, IJ
R2: AC, BE, DF, IH, GJ
and so on...

Trick is to keep these pairings and make foursomes as unique as possible.  

I took a manual stab at it and have ended up with about 5 people playing each other 3 times, rest playing twice or once.  

Thanks in advance,

Rusty

Answer

Golf pairing matrix
Hi Rusty,

I didn't try anything fancy. I set some constraints and played around with different combination in an Excel spreedsheet.

The basic requirements, as I understand them, are to
- produce a total of 26 pairings between 12 players in 5 rounds
- with as little repetition (make them as unique) as possible.

One solution I came up with is shown in the enclosed graphics (see JPEG image).

Notes:
1. The columns and rows in the matrix represent the 12 players (labeled A to L).

2. Each entry in the matrix gives a potential pairing and "round number" between two players. For example, with a "2" in row B and column D, it indicates that player B will pair-off with player D in round 2.

3. The draw is done interactively in Excel. I have used formulas in the "Violation" and "Conflict" columns to generate warnings when conditions are breached.
a) If "Conflict" has a value of 0, everything is fine.
b) If "Conflict" has a non-zero value, it indicates the round where a scheduling conflict occurs. For instance, where a player has been assigned to play with two players at the same time; this is deemed to be unreasonable.
c) If any such conflict occurs, then "Violation" will display a YES to highlight such problem.
d) To rectify such problem, the conflicting entries are deleted from the matrix and new pairings are assigned. [Since the matrix entries are symmetrical about the diagonal, only the yellow entries ever need to be changed. The entries in the bottom half will be updated automatically. This is because "E plays H" implies "H plays E".]
e) As you can see, the given solution is free of scheduling conflict.

4. The green columns "R1" through to "R5" to the right of the "Conflict" column indicates which round a player will play in. e.g., player A will play in all 5 rounds, whereas player B will "sit out" round 4. The player's opponent can easily be found from the intersecting entries in the matrix.

5. Finally, the bottom row indicates the number of playing pairs in action in each round.
Right now, I have 6 pairs playing in both round 2 and 3; 5 pairs playing in rounds 1,4 and 5. So, there will be 27 pairings in total. So, you have a bonus pairing at no cost (without creating any redundant pairings). If you want 26 pairings only, this can be easily fixed. Just cancel out the pairing between F and G in round 3 (entry shown in red).

6. Based on the pairing given in the matrix, you can almost arbitrarily put together any two playing pairs to form a group of 4.

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    CommentThank you sir, that was quite helpful. I was able to use this as a starting point and come up with a satisfactory arrangement, mostly a max of 2 repeats with a few triples, but that's okay. Appreciate your expertise and guidance. Cheers, Rusty


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Josh

Expertise

When I work through problems, I like to emphasize concepts which I believe are worth noting. I will try to answer questions in the following areas, but not at the advanced level. Algebra. Sequences & Series. Trigonometry. Functions & Graphs. Coordinate Geometry. Quadratic Polynomials. Exponential & Logarithms. Basic Calculus. Probability, Permutation and Combination. Mathematical Induction. Complex numbers. Physics problems.

Experience

I have worked as a teaching assistant in college. My hope is that more people will share knowledge without boundary, give help without seeking recognition or monetary rewards.

Education/Credentials
Bachelor degree in Engineering Science

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