Three Problems

I. The Farmer's Wheat.

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:

               Field #|1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
        --------------+-----------------------------------------------------
         First Variety|1 1 1 1 1 1 2 2 2  2  2  3  3  3  3  4  4  4  5  5  6  
        Second Variety|2 3 4 5 6 7 3 4 5  6  7  4  5  6  7  5  6  7  6  7  7

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?

II. The Schoolgirls' Walk.

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.

III. The Tournament Schedule.

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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