Flag football substitution

[microcosm]

Hoping someone can help.  I coach a youth flag football team and am looking for a way to substitute the players equally.  There are 10 players on my team and 7 players on the field.  

I know there are 120 3 digit combinations of 0-9, but I need to know what is the minimum number of combinations so that each player is used the same number of times?

Thanks in advance,

Mike

[Ian Wakeling]

Mike,

If there were 30 football matches then you could arrange it so that each player started as a substitute exactly 9 times each.  Would that be any help?

Ian.

[microcosm]

I think I figured it out.  If I have players 0-9 and substitute them like this:

012, 345, 678, 901, 234, 567, 890, 123, 456, 789

It takes 10 rotations for each player to sit out 3 times.

[Ian Wakeling]

I mentioned 30 rotations as it is possible to have round robin balance as well as balance for the number of times players sit out.  This would mean that all possible pairs will play together in the team exactly 14 times each.  With only 10 rotations this is not possble, for example in your schedule players 0 & 1 will start the match together in 6 out of the 10 matches, but the pair 3 & 6 will only start together 4 times.  I guess 10 rounds is more practical, but I can supply the 30 round schedule if you think it would be useful.

[microcosm]

I would love to see it.

Thanks,

Mike

[Ian Wakeling]

Here are the 30 rotations for the substitutes, they are arranged so that each set of three rotation contains all but one of the 10 players.

4 9 10
2 3 7
5 6 8
5 10 6
3 4 8
1 7 9
4 5 9
1 6 7
2 8 10
2 7 8
5 1 10
3 9 6
3 8 9
1 2 6
4 10 7
1 9 8
2 4 5
3 10 7
2 10 9
3 5 1
4 6 8
3 6 10
5 7 9
4 1 2
4 7 6
1 8 10
5 2 3
5 8 7
1 3 4
2 9 6

[microcosm]

Awesome.  Thanks a lot.

Mike