Mathematics 502
Homework (due Mar 21)
A. Hulpke
22) Let (X , B ) be a square 2 − (v, k, λ) design, where X = {1, . . . , v} and B = { B1 , . . . , Bv }.
Prove that there is a Latin square of order v, having the property that the set of entries
occurring in the first k rows and the ith column is Bi, j for i = 1, . . . , v.
23) Let D = (X , B ) be a t − (v, k, λ) design with x1 , . . . , xt+1 some points of X. Suppose
that µ blocks contain all these points. Use the principle of inclusion and exclusion to
show that the number of blocks containing none of x1 , . . . , xt+1 is N + (−1)t+1 µ where N
depends on t, v, k and λ only.
24) Construct an 2 − (8, 4, 6) design.
25) Find all possible parameters of non-trivial designs with 9 points.
26) Generalize the “Guess a number with lying” example from the book for a number
in {0, . . . , 31}. How many questions are minimally needed?