A mathematical conundrum

[Diable Rouge]

So, yours truly has somehow found himself organising the rota for an online quiz team - it's a squad of seven, where everyone is of a similar standard, and an 11-round season for a team of four, so somehow have to rotate in such a way that everyone gets equal game (i.e. all 7 would get 6 games, and two would get a seventh). Pen and paper has defeated me to date, but perhaps the mathematical minds of OTF will consider it a trifle.

[Walt Flanagans Dog]

Week 1	Week 2	Week 3	Week 4	Week 5	Week 6	Week 7	Week 8	Week 9	Week 10	Week 11	No. of turns
1				1	1	1	1			1	6
2	2				2	2		2		2	6
3	3	3				3		3	3	3	7
4	4	4	4			4	4		4		7
	5	5	5	5			5			5	6
		6	6	6	6			6	6		6
			7	7	7		7	7	7		6

Trial and error towards the end but it gives you an answer.

[Wouter D]

WFD's solution works, but if I were you, I'd shuffle the weeks around to make it feel more fair to the participants as the schedule unfolds. In the current schedule, persons 3 and 4 will have played three rounds before 7 played their first.

As order, I'd propose: week 1, 5, 2, 6, 3, 7, 4, 8, 9, 10, 11. Further optimizations are definitely still possible; I haven't given it a lot of thought.

[Walt Flanagans Dog]

WFD's solution works, but if I were you, I'd shuffle the weeks around to make it feel more fair to the participants as the schedule unfolds. In the current schedule, persons 3 and 4 will have played three rounds before 7 played their first.

As order, I'd propose: week 1, 5, 2, 6, 3, 7, 4, 8, 9, 10, 11. Further optimizations are definitely still possible; I haven't given it a lot of thought.

In my haste I also did a repeating set of four (weeks 1 and 7). I'm also worrying now whether 6 and 7 don't like each other but they are forced to put up with each other for 5 out of their 6 goes.

[Diable Rouge]

Through sheer luck, I managed to come up a system that rotates people every second week as best as possible:

1. 1234
2. 5671
3. 2345
4. 1673
5. 2456
6. 1374
7. 2567
8. 1342
9. 5671
10. 2345
11. 1672

[pebblethefish]

That means that your average team would be 2577 and a bit

[Capybara]

and the median is 2345

[DPDPDPDP]

I hope a question like this comes up on one of your quizzes. You’ll be right in there with the answer!

[Rogin the Armchair fan]

Why don't you find an eighth member and enter 2 teams of 4?

[Janik]

Why don't you find an eighth member and enter 2 teams of 4?

That would defeat DR's intent to mix it up and make it virtually sociable, though.

Week 1	Week 2	Week 3	Week 4	Week 5	Week 6	Week 7	Week 8	Week 9	Week 10	Week 11	No. of turns
1				1	1	1	1			1	6
2	2				2	2		2		2	6
3	3	3				3		3	3	3	7
4	4	4	4			4	4		4		7
	5	5	5	5			5			5	6
		6	6	6	6			6	6		6
			7	7	7		7	7	7		6

Trial and error towards the end but it gives you an answer.

WFD's diagonal method works well if the week 1 team contains players 1, 3, 5 and 7 rather than players 1, 2, 3 and 4. That way every quizzer shares at least one team with every other,* and no quizzer ever misses more than one week in a row (or plays more than two in a row). As a direct consequence of the later element, every quizzer will feature in one of the first two weeks, and also one of the last two.
* - some combinations do remain more likely than others. Specifically 1&3, 5&7 and 4&6. Each of these pairings happens 5 times in total. Rather than a problem this can be sold as a 'feature' - either putting couples of particularly good friends together more often than not, or (if you are feeling more competitive!) matching complimentary knowledge sets up, say a sport specialist being paired with an entertainment one. There are also pairing that happen less often than average, such as any odd number with any even number. Which, again, gives a way to 'tailor' teams whilst still maintaining overall fairness and (the impression of) neutrality.

Running that pattern, and sticking with it all the way through gives the following:-

[Janik]

Error in the teams under the table - week 6 should read 1, 3, 5, 6, not 1, 3, 5, 7. That was done manually, rather than by formulas. Hence the mistake...

[WOM]

Disappointing.