THE TOWERS OF HANOI
TREYLONTE GAITHER
PROBLEM: Find a formula for the minimum number of
moves required to complete the Towers of Hanoi with n
disks, and prove your answer to be correct.
SOLUTION: For each n Z+ , let p(n) denote the statement: The n-disk g
Zachary Scherr
1
Math 503 HW 6
Due Friday, Mar 18
Reading
1. Read sections 10.1-10.3 of Dummit and Foote
2
Problems
1. 10.1.8
Solution:
(a) The subset Tor(M ) is non-empty since 1 0 = 0 implying that 0 Tor(M ). Let x, y Tor(M )
and r R. There exist r1 , r
Zachary Scherr
1
Math 503 HW 4
Due Friday, Feb 19
Reading
1. Read sections 8.2, 8.3, 9.1, 9.2 of Dummit and Foote
2
Problems
1. DF 8.2.3
Solution: Let R be a PID and let P R be a prime ideal. As P is prime, the quotient ring R/P
will certainly be an integ
Zachary Scherr
1
Math 503 HW 7
Due Friday, Mar 25
Reading
1. Read sections 11.1, 11.2 of Dummit and Foote
2
Problems
1. Let V = cfw_f (x) Q[x] | deg(f (x) 3.
(a) Show that V is a Q-vector space, and that
B1 = cfw_1, x, x2 , x3
1
1
B2 = cfw_1, x, (x2 x),
Zachary Scherr
1
Math 503 HW 1
Due Friday, Jan 29
Reading
1. Read sections 7.1, 7.2, 7.3 of Dummit and Foote
2
Problems
1. 7.1.25
Solution:
(a) We have
=
=
(a + bi + cj + dk)(a bi cj dk)
a(a bi cj dk) + bi(a bi cj dk) + cj(a bi cj dk) + dk(a bi cj dk)
=
(
Zachary Scherr
1
Math 503 HW 8
Due Friday, Apr 8
Reading
1. Read sections 11.3, 11.4 of Dummit and Foote
2
Problems
1. Dummit and Foote 11.3.2
Solution: To solve this problem, we use the following fact. Let V be a vector space with basis
v1 , . . . , vn a
Zachary Scherr
1
Math 503 HW 3
Due Friday, Feb 12
Reading
1. Read sections 7.5, 7.6, 8.1 of Dummit and Foote
2
Problems
1. DF 7.5.2
Solution: This problem is trivial knowing how to work with universal properties. Let F denote
the quotient field of R, then
Zachary Scherr
1
Math 503 HW 10
Due Monday, Apr 25
Reading
1. Read sections 13.1, 13.2, 13.4, 13.5.
2
Problems
1. Show that g(x) = x3 + x + 1 is irreducible in F2 [x] and let be a root. Compute
1+
1 + + 2
in F2 () = cfw_a + b + c 2 | a, b, c F2 .
Solution
Zachary Scherr
1
Math 503 HW 5
Due Friday, Feb 26
Reading
1. Read Chapter 9 of Dummit and Foote
2
Problems
1. 9.1.13
Solution: We already know that if R is any commutative ring, then R[x]/(xr)
= R for any r R.
In the present situation we let R = F [y], t
Zachary Scherr
1
Math 503 HW 2
Due Friday, Feb 5
Reading
1. Read sections 7.3, 7.4, 7.5, 7.6 of Dummit and Foote
2
Problems
1. 7.1.26
Solution:
(a) First note that
(1) = (1 1) = (1) + (1)
showing that (1) = 0 and hence 1 R. From this we have,
0 = (1) = (1
Zachary Scherr
1
Math 503 HW 9
Due Friday, Apr 15
Reading
1. Read chapter 12 of Dummit and Foote and my notes on computing invariant factors.
2
Problems
1. Let F be a field.
(a) Prove that two 2 2 matrices A, B M2 (F ) are similar if and only if they have
HW 4
Due Date: 1/30 at the beginning of class.
I encourage you to work in groups, but please be sure to work on the problems on
your own, first.
The problems must be typeset in Latex.
(1) Heres something well prove in the next class: If j, k, q, r are
HW 2
Due Date: 1/23 at the beginning of class.
I encourage you to work in groups, but please be sure to work on the problems on
your own, first.
The problems must be typeset in Latex.
(1) Prove the second half of the lemma from class:
(i j)
(mod n) = [
HW 1
Due Date: 1/20 at the beginning of class.
I encourage you to work in groups, but please be sure to work on the problems on
your own, first.
The problems must be typeset in Latex.
(1) In LaTex, replicate the following exactly.
(a) Let S be an orien
HW 8RSA
Due Date: 2/8 by the beginning of class.
For this assignment, email your source code and the a print out of the sample runs
(copied and pasted into a LaTex document, e.g.).
Only one copy per group.
In this assignment youll implement RSA. In par
HW 1
Due Date: 1/20 at the beginning of class.
I encourage you to work in groups, but please be sure to work on the problems on
your own, first.
The problems must be typeset in Latex.
(1) In LaTex, replicate the following exactly.
(a) Let S be an orien
HW 2
Due Date: 1/23 at the beginning of class.
I encourage you to work in groups, but please be sure to work on the problems on
your own, first.
The problems must be typeset in Latex.
(1) Prove the second half of the lemma from class:
(i j)
(mod n) = [
HW 6Error Detecting Codes
Due Date: 2/3 at the beginning of class.
I encourage you to work in groups, but please be sure to work on the problems on
your own, first.
The problems must be typeset in Latex.
Books are identified by an International Standar
HW 5
Due Date: 2/1 at the beginning of class.
I encourage you to work in groups, but please be sure to work on the problems on
your own, first.
The problems must be typeset in Latex.
(1) F How many solutions with x between 0 and 34 are there to the sys
HW 3
Due Date: 1/25 at the beginning of class.
I encourage you to work in groups, but please be sure to work on the problems on
your own, first.
The problems must be typeset in Latex.
(1) Prove: for every a, b, c Zn , we have a n (b +n c) = a n b +n a
HW 7
Due Date: 2/6 at the beginning of class.
I encourage you to work in groups, but please be sure to work on the problems on
your own, first.
The problems must be typeset in Latex.
(1) Seven competitive math students try to share a huge hoard of stol
Solutions to the Non-book problems.
(1) Find the perpendicular distance between the parallel planes
2x 3y + 5z = 1 and 2x 3y + 5z = 4
Solution: Pick a point P , such as P = (2, 0, 1) that is in the first
plane and a point Q, such as Q = (2, 0, 0) that is
Worksheet 3
(1) Consider the function z = f (x, y) whose graph contains the point
(2, 3, 7) and which satisfies fx (2, 3) = 2, fy (2, 3) = 5.
(a) Find the gradient of f at (2, 3)
Solution. f (2, 3) =< fx (2, 3), fy (2, 3) >=< 2, 5 >
(b) In what direction
Non-book problems
(Turn in what you can on Monday, September 5th)
(1) Find the perpendicular distance between the parallel planes
2x 3y + 5z = 1 and 2x 3y + 5z = 4
(2) Describe all lines through the origin that make an angle of
3
with
the xy-plane.
(3) Fi
Non-book problems for Section 14.1
The level set of a function w = f (x, y, z) at c = where 0 is given by
(x, y, z) : x2 + y 2 + z 2 =
1
1 + 2
.
Assume the level set for f at c for any c < 0 is the empty set.
(1) Can you tell if f attains a maximum or a m
Non-book problems for Section 14.1
The level set of a function w = f (x, y, z) at c = where 0 is given by
1
(x, y, z) : x + y + z =
1 + 2
2
2
2
.
(1) Can you tell if f attains a maximum or a minimum value?
(2) As , what is happening to the level sets?
(3)
Math 211, Exam I
September 13th, 2016
1
2
3
4
5
6
7
total
14
10
12
18
16
16
14
100
Show all work and write clearly! Do as much as you can and do not assume
the problems are in order of difficulty or importance.
1. Match the descriptions of the geometric o
Department of Mathematics
Bucknell University
MATH 202
Supplementary Handout
Revised Spring 2011
Graphing Polar Functions
The purpose of this handout is to provide some further practice at sketching the graphs of
polar functions. The problems aim to devel
Worksheet 2
Consider the function z = f (x, y) whose graph contains the point (2, 3, 7)
and which satisfies fx (2, 3) = 2, fy (2, 3) = 5.
(1) Find the equation of the tangent line to the trace of f at y = 3.
What is a direction vector for the tangent line