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
|
|
|
