MAT 115H – Mathematics: An Historical Perspective
Professor Pestieau
Assignment 5 – Modern Era (18th & 19th centuries)
Due: Thursday, December 17th
Instructions
 Present your work neatly on separate sheets of paper for all the problems listed
below.
 Justify your answers, when necessary, to receive full credit.
NUMBER THEORY
Problem 1
In 1770, Joseph-Louis Lagrange proved that every natural number can be written as the sum of
four integer squares (not necessarily distinct). So, for example, 3 = 12 + 12 + 12 + 02 and 310 =
172 + 42 + 22 + 12.
a)
Write the numbers 7, 17, and 177 as the sum of four squares.
b)
Write the number 1770 as the sum of four squares.
[Bonus]
Problem 2
In 1772, Leonhard Euler came up with the following quadratic expression to generate primes:
n2 + n +41,
where n = 0, 1, 2, 3, 4, …
It can be verified that this quadratic expression returns primes for its first 40 consecutives
values, when n = 0, 1, 2,…, 39.
a)
Show that the value of this expression for n = 40 is, indeed, a composite number.
b)
Find the first 3-digit prime returned by this quadratic expression. For what value of n
does this occur?
GRAPH THEORY
Problem 3 – Garden Walks
The figure below depicts a system of bridges and land areas that a mathematician designed for a
local garden.
a)
Draw a graph that corresponds to this system of bridges and land areas.
b)
Can a person take a walk and cross each bridge exactly once? If yes, shown such a walk
in your graph. If no, explain why such a walk cannot be done.
c)
Suppose the owner of the garden accidentally burns the bridge going from H to I.
Answer the previous question now.
Problem 4 – The Knight’s Move in Chess
[Bonus]
In chess, a knight can move two squares either vertically or horizontally and then one square in
a perpendicular direction.
Depending on where the knight is situated on an 8 x 8 chessboard, he has a minimal mobility of
2 moves when he’s in a corner (check this!) and a maximal mobility of 8 moves when he’s at
least two squares away from the edge (as shown in the figure below).
Let C be the graph with 64 vertices, all corresponding to the squares of a chessboard.
Two vertices of C are joined by an edge whenever a knight can go from one of the
corresponding squares to the other in one move.
a)
Draw C.
b)
Can you find an Euler circuit in C? If no, explain why. If yes, draw this circuit.
c)
Can you find an Euler path in C? If no, explain why. If yes, draw this path.
Problem 5 – Soccer Ball
A soccer ball is in fact an Archimedean solid called a truncated icosahedron, consisting of
pentagonal and hexagonal faces (see figure above).
a)
Verify Euler’s Polyhedral Formula V + F – E = 2 for this “soccer ball” polyhedron.
b)
Draw the graph corresponding to this “soccer ball” polyhedron.
NON-EUCLIDEAN GEOMETRY
Problem 6 – A Finite Geometry
Consider a finite geometry with the six points A, B, C, D, E, and F.
In this geometry a line is defined as any two of the six points (e.g. CE or AF). Parallel lines are
defined as two lines that do not share a common point (e.g. CE is parallel to AF).
a)
How many lines are in this geometry? List them all.
b)
How many of the lines are parallel to line AB? List them all.
Suppose we now define parallel lines as lines for which the numerical equivalent of their letters
have equal sum.
For example, line BC is parallel to line AD because for both lines the numerical equivalent of
their letters add up to 5:
B+C=2+3=5
and
A + D = 1 + 4 = 5.
c)
Which lines have no other parallel lines?
d)
Which lines have exactly one other parallel line?
e)
Which lines have more than one parallel line?
Problem 7 – Fractals
[Bonus]
Myriad fractals have been created over the years by mathematicians (and others). Here is a list
of some famous ones:





The Minkowski Fractal
The Quadric Koch Curve
The 5-Point Star Fractal
Lévy’s Curve
The “H” Fractal
Pick one fractal from the list above and draw it to five stages. Use the internet, or other
sources, to see exactly how these fractals are generated.