Discrete Mathematics
Assignment 13
April 30, 2023
Problem 1.
Let p and q be the following permutations.
!
p=
1 2 3 4 5 6
2 1 3 5 4 6
!
q=
1 2 3 4 5 6
6 1 2 4 3 5
Write p and q in cycle notation.
Problem 2.
following.
Suppose p and q are the permutations defined in Problem 1. Compute each of the
2.1 p−1
2.2 q −1
2.3 p ◦ q
2.4 q ◦ p
2.5 p2
Problem 3. The order of a permutation r is the least positive integer k such that rk = i, where
i = () denotes the identity permutation. Suppose p and q are the permutations defined in Problem
1. Find the order of p and the order of q.
Problem 4.
The group of the symmetries of the hexagon consists of the following 12 symmetries.
r0 = identity
r60 = rotate 60◦ CCW
r120 = rotate 120◦ CCW
r180 = rotate 180◦ CCW
r240 = rotate 240◦ CCW
r300 = rotate 300◦ CCW
d0 = flip about the 0◦ axis
d30 = flip about the 30◦ axis
d60 = flip about the 60◦ axis
d90 = flip about the 90◦ axis
d120 = flip about the 120◦ axis
d150 = flip about the 150◦ axis
1
where CCW stands for “counterclockwise” and the flipping axes are as shown below.
120◦
150◦
90◦
60◦
30◦
0◦
Suppose we identify each corner of the hexagon by the numbers 1, 2, 3, 4, 5, 6 as shown below.
1
2
6
3
5
4
Represent each of the 12 symmetries of the hexagon listed above as a permutation on {1, 2, 3, 4, 5, 6}.
Problem 5. A shop is creating new designs of bracelets. Each bracelet is to contain 6 beads, where
two beads are red, two beads are green, and the other two beads are blue. Two bracelet designs are
considered equivalent if and only if one can be obtained from the other by rotation and/or flipping
(i.e. reflection). How many non-equivalent designs of these bracelets are there?
Problem 6. A ceramic tile company is creating a new series of square tiles. The pattern on each
tile is a 3 × 3 grid of blocks, where each of the 9 blocks is either colored gray or white. Two tile
designs are considered equivalent if and only if one can be obtained from the other by rotation.
Below are a few examples of tile designs. Observe that Tile 1 and Tile 2 are equivalent.
Tile 1
Tile 2
Tile 3
How many non-equivalent designs of these tiles are there?
————————–
2
Tile 4
Tile 5