SOLUTION: i have to arrange 56 people in groups of 7, and these groups need to change as many times as possible, but no 2 people can be in the same group more than once. how many times can

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Question 900671: i have to arrange 56 people in groups of 7, and these groups need to change as many times as possible, but no 2 people can be in the same group more than once. how many times can this happen, and what is an ordered way to have them switch?
Found 2 solutions by Edwin McCravy, AnlytcPhil:
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
Let the 56 people be numbered 1-56
Let the 8 groups be numbered 1-8 
Each group will consist of the 7 people 
in the vertical columns 

group    1  2  3  4  5  6  7  8
-------------------------------
person   1  2  3  4  5  6  7  8
person   9 10 11 12 13 14 15 16
person  17 28 19 20 21 22 23 24
person  25 26 27 28 29 30 31 32
person  33 34 35 36 37 38 39 40
person  41 42 43 44 45 46 47 48
person  49 50 51 52 53 54 55 56

To form the first change slide the people over like this,
for instance, person 9 moves from group to group 2

group    1  2  3  4  5  6  7  8
-------------------------------
person   1  2  3  4  5  6  7  8
person      9 10 11 12 13 14 15 16
person        17 28 19 20 21 22 23 24
person           25 26 27 28 29 30 31 32
person              33 34 35 36 37 38 39 40
person                 41 42 43 44 45 46 47 48
person                    49 50 51 52 53 54 55 56

Then take the people who stick out on the right and fill them
in the gaps at the first

group    1  2  3  4  5  6  7  8
-------------------------------
person   1  2  3  4  5  6  7  8
person  16  9 10 11 12 13 14 15 
person  23 24 17 28 19 20 21 22 
person  30 31 32 25 26 27 28 29
person  37 38 39 40 33 34 35 36
person  44 45 46 47 48 41 42 43 
person  51 52 53 54 55 56 49 50

For the 2nd change, do it again, and 
person 9 will now move to group 3.

group    1  2  3  4  5  6  7  8
-------------------------------
person   1  2  3  4  5  6  7  8
person     16  9 10 11 12 13 14 15 
person        23 34 17 28 19 20 21 22 
person           30 31 32 25 26 27 28 29
person              37 38 39 40 33 34 35 36
person                 44 45 46 47 48 41 42 43 
person                    51 52 53 54 55 56 49 50

group    1  2  3  4  5  6  7  8
-------------------------------
person   1  2  3  4  5  6  7  8
person  15 16  9 10 11 12 13 14 
person  21 22 23 34 17 28 19 20 
person  27 28 29 30 31 32 25 26
person  33 34 35 36 37 38 39 40
person  47 48 41 42 43 44 45 46 
person  53 54 55 56 49 50 51 52

This can continue until person 9 moves to group 8.

Counting the first time, person 9 can appear once in
each group, so that's 8 times, 7 changes.

Edwin
Answer by AnlytcPhil(1806)   (Show Source): You can put this solution on YOUR website!
Let the 56 people be numbered 1-56
Let the 8 groups be numbered 1-8 
Each group will consist of the 7 people 
in the vertical columns 

group    1  2  3  4  5  6  7  8
-------------------------------
person   1  2  3  4  5  6  7  8
person   9 10 11 12 13 14 15 16
person  17 28 19 20 21 22 23 24
person  25 26 27 28 29 30 31 32
person  33 34 35 36 37 38 39 40
person  41 42 43 44 45 46 47 48
person  49 50 51 52 53 54 55 56

To form the first change slide the people over like this,
for instance, person 9 moves from group to group 2

group    1  2  3  4  5  6  7  8
-------------------------------
person   1  2  3  4  5  6  7  8
person      9 10 11 12 13 14 15 16
person        17 28 19 20 21 22 23 24
person           25 26 27 28 29 30 31 32
person              33 34 35 36 37 38 39 40
person                 41 42 43 44 45 46 47 48
person                    49 50 51 52 53 54 55 56

Then take the people who stick out on the right and fill them
in the gaps at the first

group    1  2  3  4  5  6  7  8
-------------------------------
person   1  2  3  4  5  6  7  8
person  16  9 10 11 12 13 14 15 
person  23 24 17 28 19 20 21 22 
person  30 31 32 25 26 27 28 29
person  37 38 39 40 33 34 35 36
person  44 45 46 47 48 41 42 43 
person  51 52 53 54 55 56 49 50

For the 2nd change, do it again, and 
person 9 will now move to group 3.

group    1  2  3  4  5  6  7  8
-------------------------------
person   1  2  3  4  5  6  7  8
person     16  9 10 11 12 13 14 15 
person        23 34 17 28 19 20 21 22 
person           30 31 32 25 26 27 28 29
person              37 38 39 40 33 34 35 36
person                 44 45 46 47 48 41 42 43 
person                    51 52 53 54 55 56 49 50

group    1  2  3  4  5  6  7  8
-------------------------------
person   1  2  3  4  5  6  7  8
person  15 16  9 10 11 12 13 14 
person  21 22 23 34 17 28 19 20 
person  27 28 29 30 31 32 25 26
person  33 34 35 36 37 38 39 40
person  47 48 41 42 43 44 45 46 
person  53 54 55 56 49 50 51 52

This can continue until person 9 moves to group 8.

Counting the first time, person 9 can appear once in
each group, so that 8 times, 7 changes.

Edwin
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