Combinatorics - MATH 0345
Exam 3
May 18, 2009
Name:
Honor Code Pledge
Signature
Directions: Please complete all but one problem.
1. Prove that the addition table of Z4is a Latin square without an orthogonal mate.
2. How many 2 −by −nLatin rectangles have first row equal to
0 1 2 · · · n−1 ?
Estimate how many 3 −by −nrectangles have this as their first row.
3. Solve the recurrence relation hn=−hn−1+ 1,(n≥1); h0= 0,by examining the first
few values for a formula and then proving your conjectured formula by induction.
4. Call a subset Sof the integers {1,2, . . . , n}extraordinary provided its smallest integer
equals its size:
min{x:x∈S}=|S|.
For example, S={3,7,8}is extraordinary. Let gnbe the number of extraordinary
subsets of {1,2, . . . , n}. Determine a linear recurrence relation for gn.
5. Let p1≤p2≤. . . ≤pkbe primes and n=p1×p2×. . . ×pk. Prove that N(n), the
largest number of MOLS of order n, is at least p1−1.
6. Solve the recurrence relation hn= 8hn−1−16hn−2,(n≥2) with initial value h0=−1
and h1= 0.
1