# Combination/Permutation Problem
**URL:** https://boards.straightdope.com/t/combination-permutation-problem/740566
**Category:** Factual Questions
**Created:** [December 15, 2015, 9:40pm UTC](https://boards.straightdope.com/t/combination-permutation-problem/740566 "2015-12-15T21:40:36Z")
**Posts on this page:** 7
**Page:** 1
### Author:  [@notfrommensa](https://boards.straightdope.com/u/notfrommensa)
#### Post date: [December 15, 2015, 9:40pm UTC](https://boards.straightdope.com/t/combination-permutation-problem/740566/1 "2015-12-15T21:40:36Z")
I have a population of 28 golfers going on a 4 day golf trip.
Other than brute force is there an easy way to line up these golfers in seven foursomes per day so that no player plays with any other player more than once.
---
### Author:  [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)
#### Post date: [December 15, 2015, 10:05pm UTC](https://boards.straightdope.com/t/combination-permutation-problem/740566/2 "2015-12-15T22:05:56Z")
It helps that 7 is prime…
Day 1
```auto
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
21 22 23 24
25 26 27 28
```
Day 2
```auto
1 6 11 16
5 10 15 20
9 14 19 24
13 18 23 28
17 22 27 4
21 26 3 8
25 2 7 12
```
Day 3
```auto
1 10 19 28
5 14 23 4
9 18 27 8
13 22 3 12
17 26 7 16
21 30 11 20
25 34 15 24
```
Day 4
```auto
1 14 27 12
5 18 3 16
9 22 7 20
13 26 11 24
17 2 15 28
21 6 19 4
25 10 23 8
```
I think it should be relatively straightforward what’s going on here, but I’ll answer any questions you have. You could even extend the trip by a few days.
---
### Author:  [@notfrommensa](https://boards.straightdope.com/u/notfrommensa)
#### Post date: [December 15, 2015, 10:19pm UTC](https://boards.straightdope.com/t/combination-permutation-problem/740566/3 "2015-12-15T22:19:35Z")
Ok Thanks, Yes, I see that you are going diagonally on each subsequent day.
---
### Author:  [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)
#### Post date: [December 15, 2015, 10:35pm UTC](https://boards.straightdope.com/t/combination-permutation-problem/740566/4 "2015-12-15T22:35:41Z")
I like to think of each column as a wheel on a suit case style lock. The first wheel is fixed, the second wheel moves one spot each day, the third two spots, and the fourth three spots. Since 1, 2, 3 are all relatively prime to 7 we don’t run into any trouble. The same technique would not work without modification would not work for a a group of 24 golfers for this reason.
---
### Author:  [@Novelty\_Bobble](https://boards.straightdope.com/u/Novelty_Bobble)
#### Post date: [December 15, 2015, 11:06pm UTC](https://boards.straightdope.com/t/combination-permutation-problem/740566/5 "2015-12-15T23:06:14Z")
I may be going a little mad here, but on day 3 why have you suddenly lost players 2 and 6? and where have 30 and 34 come from? Are you playing ringers here?
---
### Author:  [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)
#### Post date: [December 15, 2015, 11:16pm UTC](https://boards.straightdope.com/t/combination-permutation-problem/740566/6 "2015-12-15T23:16:43Z")
Oops.
Those players are equivalent modulo 28.
Day 3 (corrected)
```auto
1 10 19 28
5 14 23 4
9 18 27 8
13 22 3 12
17 26 7 16
21 2 11 20
25 6 15 24
```
---
### Author:  [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)
#### Post date: [December 16, 2015, 1:01am UTC](https://boards.straightdope.com/t/combination-permutation-problem/740566/7 "2015-12-16T01:01:17Z")
This is a resolvable balanced incomplete block design where v = 28, k = 4, and λ = 1:
> **[Combinatorial design](https://en.wikipedia.org/wiki/Combinatorial_design)**
>
> Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangemen...