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RobPratt
RobPratt
  • 15.7k
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  • 35
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You can use up all 150 slots, with exactlyeither 7 or 8 people per team per day, as follows:

Day 1:
Team 1 {1,2,3,4,5,6,7,8,9,10}
Team 2 {1,2,3,4,5,6,78,910}
Team 3 {1,2,3,4,5,6,7,89,10}
Team 4 {1,2,3,4,6,78,109}
Team 5 {1,2,3,4,5,6,7,8,9}
Team 6 {1,2,3,4,5,6,78,9,10}
Team 7 {1,2,3,4,5,6,7,108,9}
Team 8 {1,2,3,4,6,97,8,10}
Team 9 {1,2,3,4,5,6,7,108,9}
Team 10 {1,2,3,4,5,6,7,108,9}
Team 11 {1,2,3,4,5,6,7,8,10}
Team 12 {1,2,3,4,5,6,7,8,9,10}
Team 13 {21,32,4,5,6,7,9,10}
Team 14 {1,2,3,45,6,7,8,9,10}
Team 15 {1,2,3,4,56,7,9,10}
Team 16 {1,2,4,5,6,7,8,9,10}
Team 17 {1,2,3,5,6,7,9,10}
Team 18 {1,2,3,4,5,6,7,8,10}
Team 19 {1,2,3,4,5,6,7,8,10}
Team 20 {1,2,3,4,5,6,7,8,9,10}

Day 2:
Team 1 {21,3,4,5,6,7,8,9,10}
Team 2 {1,2,3,4,5,7,9,10}
Team 3 {1,2,3,4,5,6,7,8,9,10}
Team 4 {1,2,3,4,5,6,7,9,10}
Team 5 {1,2,3,4,5,6,7,8,9}
Team 6 {1,2,3,4,5,6,78,910}
Team 7 {1,2,3,4,5,7,8,10}
Team 8 {1,2,3,4,5,68,9,10}
Team 9 {1,2,3,4,5,6,79,10}
Team 10 {1,2,3,4,5,6,7,8,9,10}
Team 11 {1,2,3,4,5,76,8,9,10}
Team 12 {1,2,3,4,5,6,7,8,9}
Team 13 {1,2,3,4,5,6,7,910}
Team 14 {1,2,3,4,5,6,7,8,9}
Team 15 {1,2,3,4,5,6,7,8}
Team 16 {1,2,3,4,5,6,7,8,9,10}
Team 17 {1,2,3,4,5,6,78,9,10}
Team 18 {1,2,3,4,5,6,78,109}
Team 19 {1,2,3,4,5,6,7,89}
Team 20 {1,2,3,4,5,6,7,8,10}

Day 3:
Team 1 {1,2,3,4,56,7,9,10}
Team 2 {1,2,3,4,5,6,7,98,10}
Team 3 {1,2,3,4,5,6,7,8,9}
Team 4 {1,2,3,4,5,6,79,10}
Team 5 {1,2,3,4,5,6,79,810}
Team 6 {1,2,3,5,6,7,9,10}
Team 7 {1,2,3,4,5,6,7,108}
Team 8 {21,32,4,5,67,98,109}
Team 9 {1,2,3,4,5,6,7,8,10}
Team 10 {1,2,3,4,5,6,78,10}
Team 11 {1,2,3,4,5,6,7,98,10}
Team 12 {1,2,3,4,5,6,7,8,9,10}
Team 13 {1,2,3,4,5,6,78,9}
Team 14 {1,2,3,4,5,6,7,910}
Team 15 {1,2,3,4,5,6,7,8,10}
Team 16 {1,3,4,5,6,7,8,9,10}
Team 17 {21,32,4,5,6,7,98,10}
Team 18 {1,2,3,4,5,6,7,8,9,10}
Team 19 {21,3,4,5,6,7,8,10}
Team 20 {1,2,3,4,5,6,78,9,10}

Day 4:
Team 1 {1,2,3,4,5,6,7,9,10}
Team 2 {1,2,3,4,5,6,7,8,9}
Team 3 {1,2,3,4,5,6,7,8,10}
Team 4 {1,2,3,4,5,6,7,8,9,10}
Team 5 {1,2,3,4,5,6,7,89,10}
Team 6 {1,2,3,4,5,6,7,8,9,10}
Team 7 {1,3,42,5,6,7,8,9,10}
Team 8 {1,2,5,6,7,8,9,10}
Team 9 {1,3,4,5,6,7,8,9,10}
Team 10 {1,2,3,4,5,6,7,9,10}
Team 11 {1,2,4,5,6,7,8,9,10}
Team 12 {1,2,3,4,5,6,7,8,9,10}
Team 13 {1,2,3,4,56,7,8,9,10}
Team 14 {1,2,3,4,5,6,7,98,10}
Team 15 {1,32,4,5,6,7,8,9,10}
Team 16 {1,2,3,4,5,6,7,89}
Team 17 {1,2,3,4,5,6,7,910}
Team 18 {1,2,3,4,5,6,7,108}
Team 19 {1,2,3,4,5,6,7,8,9,10}
Team 20 {1,2,3,4,5,6,7,9,10}

Day 5:
Team 1 {1,2,3,4,5,7,98,10}
Team 2 {1,2,3,4,5,6,7,8,9}
Team 3 {1,2,3,4,5,6,7,8,10}
Team 4 {1,2,3,4,57,68,79,10}
Team 5 {1,2,43,5,6,7,8,9,10}
Team 6 {21,3,4,5,6,7,9,10}
Team 7 {1,2,3,4,5,6,79,10}
Team 8 {1,2,3,4,5,6,97,8}
Team 9 {1,2,3,4,5,6,7,10}
Team 10 {1,2,3,4,5,6,78,109}
Team 11 {1,2,3,4,5,6,7,8,9,10}
Team 12 {1,2,4,5,6,7,8,9,10}
Team 13 {1,2,3,4,5,6,7,98}
Team 14 {1,2,3,4,5,76,8,9,10}
Team 15 {1,2,4,5,6,7,8,9,10}
Team 16 {1,2,3,4,5,6,7,8,9,10}
Team 17 {1,2,3,4,5,6,7,8,9,10}
Team 18 {1,2,3,4,5,6,7,9,10}
Team 19 {1,2,3,4,5,6,7,9,10}
Team 20 {1,2,3,4,5,6,78,109}

I obtained the solution via integer linear programming, and 8$8-7=1$ is the minimum possible valuerange of the largest team member counts per day.

You can use up all 150 slots, with exactly 8 people per team, as follows:

Day 1:
Team 1 {1,2,3,4,5,6,7,9}
Team 2 {1,2,3,4,5,6,7,9}
Team 3 {1,2,3,4,5,6,7,8}
Team 4 {1,2,4,6,7,10}
Team 5 {1,2,3,4,5,6,7,8}
Team 6 {1,2,3,4,5,6,7,9}
Team 7 {1,2,3,4,5,6,7,10}
Team 8 {1,2,3,4,6,9,10}
Team 9 {1,2,3,4,5,6,7,10}
Team 10 {1,2,3,4,5,6,7,10}
Team 11 {1,2,3,4,5,6,7}
Team 12 {1,2,3,4,5,6,7,8}
Team 13 {2,3,4,5,6,7,9}
Team 14 {1,2,3,4,6,7,9}
Team 15 {1,2,3,4,5,7}
Team 16 {1,2,4,5,6,7,8}
Team 17 {1,2,3,5,6,7,9}
Team 18 {1,2,3,4,5,6,7,10}
Team 19 {1,2,3,4,5,6,7,8}
Team 20 {1,2,3,4,5,6,7,10}

Day 2:
Team 1 {2,3,4,5,6,7,9}
Team 2 {1,2,3,5,7,9}
Team 3 {1,2,3,4,5,6,7,8}
Team 4 {1,2,3,4,5,6,7,10}
Team 5 {1,2,3,4,5,7,8}
Team 6 {1,2,3,4,6,7,9}
Team 7 {1,2,3,5,7,10}
Team 8 {1,2,3,4,5,6,9,10}
Team 9 {1,2,3,4,5,6,7,10}
Team 10 {1,2,3,4,5,6,7,10}
Team 11 {1,2,3,4,5,7,9}
Team 12 {1,2,3,4,5,6,7,8}
Team 13 {1,2,3,4,5,7,9}
Team 14 {1,2,3,4,5,6,7,9}
Team 15 {1,2,3,4,5,7,8}
Team 16 {1,2,3,4,5,6,7,8}
Team 17 {1,2,3,4,5,6,7,9}
Team 18 {1,2,3,4,5,6,7,10}
Team 19 {1,2,3,4,5,6,7,8}
Team 20 {1,2,3,4,5,6,7,10}

Day 3:
Team 1 {1,2,3,4,5,7,9}
Team 2 {1,2,3,4,5,6,7,9}
Team 3 {1,2,3,4,5,6,7,8}
Team 4 {1,2,3,4,5,6,7,10}
Team 5 {1,2,3,4,5,6,7,8}
Team 6 {1,2,3,5,6,7,9}
Team 7 {1,3,4,5,6,7,10}
Team 8 {2,3,4,5,6,9,10}
Team 9 {1,2,3,4,5,7,10}
Team 10 {1,2,3,4,5,6,7,10}
Team 11 {1,2,3,4,5,7,9}
Team 12 {1,2,3,4,5,6,7,8}
Team 13 {1,2,3,4,5,6,7,9}
Team 14 {1,2,3,4,5,6,7,9}
Team 15 {1,3,4,5,6,7,8}
Team 16 {1,3,4,5,6,7,8}
Team 17 {2,3,4,5,6,7,9}
Team 18 {1,2,3,4,5,6,7,10}
Team 19 {2,3,4,5,6,7,8}
Team 20 {1,2,3,4,5,6,7,10}

Day 4:
Team 1 {1,2,3,4,5,6,7,9}
Team 2 {1,2,3,4,5,6,7,9}
Team 3 {1,2,3,4,5,6,7,8}
Team 4 {1,2,3,4,5,6,7,10}
Team 5 {1,2,3,4,5,6,7,8}
Team 6 {1,2,3,4,5,6,7,9}
Team 7 {1,3,4,5,6,7,10}
Team 8 {1,2,5,6,9,10}
Team 9 {3,4,5,6,7}
Team 10 {1,2,3,4,5,6,7,10}
Team 11 {1,2,4,5,6,7,9}
Team 12 {1,2,3,4,5,6,7,8}
Team 13 {1,2,3,4,5,7,9}
Team 14 {1,2,4,5,6,7,9}
Team 15 {1,3,4,5,6,7,8}
Team 16 {1,2,3,4,5,6,7,8}
Team 17 {1,2,3,4,5,6,7,9}
Team 18 {1,2,3,4,5,6,7,10}
Team 19 {1,2,3,4,5,6,7,8}
Team 20 {1,2,3,4,5,6,7,10}

Day 5:
Team 1 {1,3,4,5,7,9}
Team 2 {1,2,3,4,5,6,7,9}
Team 3 {1,2,3,4,5,6,7,8}
Team 4 {1,2,3,4,5,6,7,10}
Team 5 {1,2,4,5,6,7,8}
Team 6 {2,3,4,5,6,7,9}
Team 7 {1,2,3,4,5,6,7,10}
Team 8 {1,2,3,4,5,6,9}
Team 9 {1,2,3,4,5,6,7,10}
Team 10 {2,3,4,5,6,7,10}
Team 11 {1,2,3,4,5,6,7,9}
Team 12 {1,2,4,5,6,7,8}
Team 13 {1,2,3,4,5,6,7,9}
Team 14 {1,2,3,4,5,7,9}
Team 15 {1,2,4,5,6,7,8}
Team 16 {1,2,3,4,5,6,7,8}
Team 17 {1,2,3,4,5,6,7,9}
Team 18 {1,2,3,4,5,6,7,10}
Team 19 {1,2,3,4,5,6,7}
Team 20 {1,2,3,4,5,6,7,10}

I obtained the solution via integer linear programming, and 8 is the minimum possible value of the largest team.

You can use up all 150 slots, with either 7 or 8 people per team per day, as follows:

Day 1:
Team 1 {1,2,5,6,7,8,9,10}
Team 2 {1,2,3,4,5,6,8,10}
Team 3 {1,2,4,6,7,9,10}
Team 4 {1,2,3,4,6,8,9}
Team 5 {1,2,3,4,5,7,8,9}
Team 6 {2,3,4,5,6,8,9,10}
Team 7 {1,2,3,5,6,7,8,9}
Team 8 {1,3,4,6,7,8,10}
Team 9 {1,3,4,5,6,7,8,9}
Team 10 {1,3,4,5,7,8,9}
Team 11 {1,3,4,5,6,7,8,10}
Team 12 {3,4,5,6,7,8,9,10}
Team 13 {1,2,4,6,7,9,10}
Team 14 {2,3,5,6,7,8,9,10}
Team 15 {1,2,3,6,7,9,10}
Team 16 {1,4,5,7,8,9,10}
Team 17 {1,2,3,5,6,7,9,10}
Team 18 {3,4,5,6,7,8,10}
Team 19 {2,3,4,5,7,8,10}
Team 20 {1,2,5,7,8,9,10}

Day 2:
Team 1 {1,3,5,6,7,8,9,10}
Team 2 {1,3,4,5,7,9,10}
Team 3 {1,3,4,7,8,9,10}
Team 4 {3,4,5,6,7,9,10}
Team 5 {3,4,5,6,7,8,9}
Team 6 {1,2,3,4,5,6,8,10}
Team 7 {1,2,3,4,5,7,8,10}
Team 8 {1,3,4,5,8,9,10}
Team 9 {1,2,4,5,6,9,10}
Team 10 {1,4,5,6,7,8,9,10}
Team 11 {2,3,5,6,8,9,10}
Team 12 {1,2,3,4,6,7,8,9}
Team 13 {1,2,4,5,6,7,10}
Team 14 {1,3,4,5,6,7,8,9}
Team 15 {1,2,3,4,5,6,7,8}
Team 16 {1,2,3,4,7,8,9,10}
Team 17 {1,2,4,5,6,8,9,10}
Team 18 {1,2,3,4,5,8,9}
Team 19 {1,2,3,4,6,7,9}
Team 20 {1,2,3,4,6,7,8,10}

Day 3:
Team 1 {1,2,3,4,6,7,9,10}
Team 2 {1,3,5,6,7,8,10}
Team 3 {1,3,4,5,7,8,9}
Team 4 {1,2,3,4,5,6,9,10}
Team 5 {1,2,4,5,6,9,10}
Team 6 {1,2,3,5,6,7,9,10}
Team 7 {1,2,3,4,5,6,8}
Team 8 {1,2,4,5,7,8,9}
Team 9 {2,3,5,6,7,8,10}
Team 10 {1,2,4,5,6,8,10}
Team 11 {1,3,4,5,6,7,8,10}
Team 12 {1,2,3,5,7,8,9,10}
Team 13 {1,2,3,4,5,6,8,9}
Team 14 {1,2,3,4,5,6,7,10}
Team 15 {1,2,3,4,6,8,10}
Team 16 {1,3,4,6,7,8,9,10}
Team 17 {1,2,4,5,7,8,10}
Team 18 {1,4,5,6,7,8,9,10}
Team 19 {1,3,5,6,7,8,10}
Team 20 {1,3,4,5,6,8,9,10}

Day 4:
Team 1 {1,2,3,4,5,6,9,10}
Team 2 {2,3,4,6,7,8,9}
Team 3 {1,2,5,6,7,8,10}
Team 4 {1,4,5,6,7,8,9,10}
Team 5 {1,3,4,5,6,7,9,10}
Team 6 {1,2,3,7,8,9,10}
Team 7 {1,2,5,6,7,8,9,10}
Team 8 {2,5,6,7,8,9,10}
Team 9 {1,3,4,5,7,8,9,10}
Team 10 {1,2,3,4,7,9,10}
Team 11 {1,2,4,7,8,9,10}
Team 12 {1,2,3,4,6,8,9,10}
Team 13 {3,4,6,7,8,9,10}
Team 14 {1,2,3,4,5,7,8,10}
Team 15 {1,2,4,5,7,8,9,10}
Team 16 {1,2,3,4,5,6,9}
Team 17 {1,3,4,5,6,7,10}
Team 18 {2,3,4,5,6,7,8}
Team 19 {2,4,5,6,7,8,9,10}
Team 20 {1,2,4,5,6,7,9,10}

Day 5:
Team 1 {1,2,3,4,5,7,8,10}
Team 2 {1,2,3,5,6,7,8,9}
Team 3 {2,3,4,5,6,7,8,10}
Team 4 {2,3,4,7,8,9,10}
Team 5 {2,3,5,6,8,9,10}
Team 6 {1,3,4,5,6,7,9,10}
Team 7 {1,3,4,5,6,9,10}
Team 8 {1,2,3,4,5,6,7,8}
Team 9 {1,2,4,5,6,7,10}
Team 10 {1,2,3,5,6,8,9}
Team 11 {1,2,5,6,7,8,9,10}
Team 12 {1,2,5,6,7,8,9,10}
Team 13 {1,2,3,5,6,7,8}
Team 14 {2,3,4,5,6,8,9,10}
Team 15 {1,2,5,6,7,8,9,10}
Team 16 {1,2,5,6,8,9,10}
Team 17 {1,2,3,5,7,8,9,10}
Team 18 {1,2,3,5,7,9,10}
Team 19 {1,2,3,5,7,9,10}
Team 20 {1,2,3,5,6,8,9}

I obtained the solution via integer linear programming, and $8-7=1$ is the minimum range of team member counts per day.

Source Link
RobPratt
RobPratt
  • 15.7k
  • 1
  • 35
  • 61

You can use up all 150 slots, with exactly 8 people per team, as follows:

Day 1:
Team 1 {1,2,3,4,5,6,7,9}
Team 2 {1,2,3,4,5,6,7,9}
Team 3 {1,2,3,4,5,6,7,8}
Team 4 {1,2,4,6,7,10}
Team 5 {1,2,3,4,5,6,7,8}
Team 6 {1,2,3,4,5,6,7,9}
Team 7 {1,2,3,4,5,6,7,10}
Team 8 {1,2,3,4,6,9,10}
Team 9 {1,2,3,4,5,6,7,10}
Team 10 {1,2,3,4,5,6,7,10}
Team 11 {1,2,3,4,5,6,7}
Team 12 {1,2,3,4,5,6,7,8}
Team 13 {2,3,4,5,6,7,9}
Team 14 {1,2,3,4,6,7,9}
Team 15 {1,2,3,4,5,7}
Team 16 {1,2,4,5,6,7,8}
Team 17 {1,2,3,5,6,7,9}
Team 18 {1,2,3,4,5,6,7,10}
Team 19 {1,2,3,4,5,6,7,8}
Team 20 {1,2,3,4,5,6,7,10}

Day 2:
Team 1 {2,3,4,5,6,7,9}
Team 2 {1,2,3,5,7,9}
Team 3 {1,2,3,4,5,6,7,8}
Team 4 {1,2,3,4,5,6,7,10}
Team 5 {1,2,3,4,5,7,8}
Team 6 {1,2,3,4,6,7,9}
Team 7 {1,2,3,5,7,10}
Team 8 {1,2,3,4,5,6,9,10}
Team 9 {1,2,3,4,5,6,7,10}
Team 10 {1,2,3,4,5,6,7,10}
Team 11 {1,2,3,4,5,7,9}
Team 12 {1,2,3,4,5,6,7,8}
Team 13 {1,2,3,4,5,7,9}
Team 14 {1,2,3,4,5,6,7,9}
Team 15 {1,2,3,4,5,7,8}
Team 16 {1,2,3,4,5,6,7,8}
Team 17 {1,2,3,4,5,6,7,9}
Team 18 {1,2,3,4,5,6,7,10}
Team 19 {1,2,3,4,5,6,7,8}
Team 20 {1,2,3,4,5,6,7,10}

Day 3:
Team 1 {1,2,3,4,5,7,9}
Team 2 {1,2,3,4,5,6,7,9}
Team 3 {1,2,3,4,5,6,7,8}
Team 4 {1,2,3,4,5,6,7,10}
Team 5 {1,2,3,4,5,6,7,8}
Team 6 {1,2,3,5,6,7,9}
Team 7 {1,3,4,5,6,7,10}
Team 8 {2,3,4,5,6,9,10}
Team 9 {1,2,3,4,5,7,10}
Team 10 {1,2,3,4,5,6,7,10}
Team 11 {1,2,3,4,5,7,9}
Team 12 {1,2,3,4,5,6,7,8}
Team 13 {1,2,3,4,5,6,7,9}
Team 14 {1,2,3,4,5,6,7,9}
Team 15 {1,3,4,5,6,7,8}
Team 16 {1,3,4,5,6,7,8}
Team 17 {2,3,4,5,6,7,9}
Team 18 {1,2,3,4,5,6,7,10}
Team 19 {2,3,4,5,6,7,8}
Team 20 {1,2,3,4,5,6,7,10}

Day 4:
Team 1 {1,2,3,4,5,6,7,9}
Team 2 {1,2,3,4,5,6,7,9}
Team 3 {1,2,3,4,5,6,7,8}
Team 4 {1,2,3,4,5,6,7,10}
Team 5 {1,2,3,4,5,6,7,8}
Team 6 {1,2,3,4,5,6,7,9}
Team 7 {1,3,4,5,6,7,10}
Team 8 {1,2,5,6,9,10}
Team 9 {3,4,5,6,7}
Team 10 {1,2,3,4,5,6,7,10}
Team 11 {1,2,4,5,6,7,9}
Team 12 {1,2,3,4,5,6,7,8}
Team 13 {1,2,3,4,5,7,9}
Team 14 {1,2,4,5,6,7,9}
Team 15 {1,3,4,5,6,7,8}
Team 16 {1,2,3,4,5,6,7,8}
Team 17 {1,2,3,4,5,6,7,9}
Team 18 {1,2,3,4,5,6,7,10}
Team 19 {1,2,3,4,5,6,7,8}
Team 20 {1,2,3,4,5,6,7,10}

Day 5:
Team 1 {1,3,4,5,7,9}
Team 2 {1,2,3,4,5,6,7,9}
Team 3 {1,2,3,4,5,6,7,8}
Team 4 {1,2,3,4,5,6,7,10}
Team 5 {1,2,4,5,6,7,8}
Team 6 {2,3,4,5,6,7,9}
Team 7 {1,2,3,4,5,6,7,10}
Team 8 {1,2,3,4,5,6,9}
Team 9 {1,2,3,4,5,6,7,10}
Team 10 {2,3,4,5,6,7,10}
Team 11 {1,2,3,4,5,6,7,9}
Team 12 {1,2,4,5,6,7,8}
Team 13 {1,2,3,4,5,6,7,9}
Team 14 {1,2,3,4,5,7,9}
Team 15 {1,2,4,5,6,7,8}
Team 16 {1,2,3,4,5,6,7,8}
Team 17 {1,2,3,4,5,6,7,9}
Team 18 {1,2,3,4,5,6,7,10}
Team 19 {1,2,3,4,5,6,7}
Team 20 {1,2,3,4,5,6,7,10}

I obtained the solution via integer linear programming, and 8 is the minimum possible value of the largest team.