S15 Exam 2 Problem 1
Combinatorics, Dave Bayer
[Reserved for Score]
exam01a2p1
8 6 6 0
Exam 01
Name
Uni
[1] Up to rotational symmetry, how many nine bead necklaces have three red beads and six blue beads?
S15 Exam 2 Problem 2
Combinatorics, Dave Bayer
[Reserved for Score]
exam01a2p2
8 8 1 5
Exam 01
[2] Up to symmetry (rotations and flips), how many ways can the squares of a 4 by 4 checkerboard be
colored using n colors?
S15 Exam 2 Problem 3
Combinatorics, Dave Bayer
[Reserved for Score]
exam01a2p3
1 9 6 5
Exam 01
[3] Up to rotational symmetry, how many ways can the eight corners of a cube be colored using n colors?
S15 Exam 2 Problem 4
Combinatorics, Dave Bayer
[Reserved for Score]
exam01a2p4
3 8 4 5
Exam 01
[4] Give a proof of Burnside’s Lemma: If a group G acts on a set of patterns X, then the number of distinct
patterns up to symmetry is equal to the average number of patterns fixed by an element of the group:
1 X g
|X |
|G| g∈G
S15 Exam 2 Problem 5
Combinatorics, Dave Bayer
[Reserved for Score]
exam01a2p5
7 3 8 7
Exam 01
[5] Up to symmetry (rotations and flips), how many ways can one mark six out of the 24 triangles of the
following figure?