udosuk wrote:Think smaller...

The library has 7 staff members. Each day of the week (Monday to Sunday) 3 members are rostered to work. Try to work out a weekly schedule that each member works 3 days a week and is teamed up with every other colleague once a week...

This looks like projective geometry.

The axioms for a projective plane are:

A. On any two distinct points there is exactly one line.

B. On any two distinct lines there is exactly one point.

C. On each line there are at least 3 points.

D. Not all points are on the same line.

(Note: For point P to be on line L means the same thing as for line L to be on point P.)

If there are N points on each line, then there are N lines on each point, and N^2 - N + 1 points (and this same number of lines) altogether.

If N=3, one can construct a projective plane as follows. The points are: (1) the three vertices of an equilateral triangle, (2) the mid-points of the three sides, and (3) the center of the triangle. The lines are: (1) the three sides, (2) the three medians (joining the mid-point of each side to the opposite vertex), and (3) the inscribed circle.

Now just let each point be a library employee, and each line be a day of the week (or vice versa), and you have your solution.

Bill Smythe