Eleven theater groups took part in a festival. Every day some of the groups put on their plays while the others watched. When the festival was over it was possible to affirm that each one of the groups had been able to attend, at least once, the performances of other groups. What is the minimum number of days that the festival lasted?
#23712. Asked by William. (Oct 30 02 3:18 PM)
The Festival lasted at least 6 days. |
Let the 11 groups be A1, A2, A3, ..., A11. If the 11 groups put on their plays according to following program of events, it is easy to see how each group was able to be present at least one of the performances of every other group.
GROUP DAYS ON WHICH THEY PERFORMED
A1 1 and 2
A2 1 and 3
A3 1 and 4
A4 1 and 5
A5 2 and 3
A6 2 and 4
A7 2 and 5
A8 3 and 4
A9 3 and 5
A10 4 and 5
It can be shown that a similar program of events allowing each group to attend at least one of all the other groups' performances would not have been possible in five days.