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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?