Combinatorics - MATH 0345
Exam 3
December 18, 2006
Name:
Honor Code Pledge
Signature
Directions: Please complete all but 1 problem. If you complete all seven problems, I
will count your best six.
1. Solve the following recurrence relation by examining the first few values for a formula
and then proving your conjectured formula by induction.
hn= 2hn−1+ 1,(n≥1); h0= 1
2. Solve the nonhomogeneous recurrence relation
hn= 4hn−1+ 4n,(n≥1)
h0= 3
3. Construct 2 mutually orthogonal latin squares (MOLS) of order 9.
4. A Latin Square Abased on Znof order nis symmetric, provided the entry aij at
row i, column jequals the entry aji at column j, row ifor all i6=j. It is idempotent
provided that its entries on the diagonal running from upper left to lower right are
0,1,2, . . . , n −1. Prove that a symmetric, idempotent Latin square has odd order
and construct an example of order 5.
5. •Does there exist a BIBD with parameters b= 10, v = 8, r = 5 and k= 4?
Justify your answer.
•True or false: In a BIBD, if k= 3, then either λis even or vis odd. (Prove or
give a counterexample.)
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