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March 1997 Presentation Topic (continued)

# Three Problems

A farmer wants to plant seven
different varieties of wheat. He has many fields available, and can
plant more than one variety in each (but wants the
same number in each). *For every pair
of varieties, there needs to be exactly one field in which they're both
planted* (so he can compare their relative yields under identical
soil conditions, without wasting more than one field per comparison).
He can't plant all seven in one field; it's
too many for a single field, and doesn't give him
a chance to see each variety growing in several different soil conditions.
How many fields must he use?
One solution is to plant two varieties in each field, using 21 fields in
total (there are 21 different pairs of varieties, each needing a field).
This solution is tabulated below:

However, using that many fields would be expensive!

There is one solution that uses fewer fields--in fact, only one. It is called the
*Fano Plane*, because
it has an interesting interpretation as a kind of geometry.
Can you find it?

In an all-girls boarding school, there are 9 girls. On four days during
the week they go out for a walk in three lines of three girls each.
Find a way to schedule their four walks in such a way that each girl
has every other girl as a line-mate exactly once.
In a certain sports tournament, each team must play every other team
exactly once. No team can play twice in the same day. This means each
team must play on *n*-1 different days
(where *n* is the number of teams)--one day for each opponent.
The ideal tournament would be if every team plays every day; then the
whole tournament takes only *n*-1 days (otherwise it would take longer).
Can this be done? Can you find explicit schedules for the
*n*=4, *n*=6, and *n*=8 cases?

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