Challenges
Computational Questions
Calculus
(5 pts) Write a program that plots the polynomial (x-1)^2 + (x+1)^2.
(10 pts) Write a program that plots the derivative of that polynomial.
(15 pts) Calculate the area under the curve between x=-1 and x=1.
Linear Algebra
(5 pts) Solve the system of three equations: (1) 2x + 2y + z = 11, (2) 2x + y + z = 8, (3) x + y + 2z = 7.
(5 pts) Write a coefficient matrix of the above system.
Graph Theory
(10 pts) Draw a graph represented by the matrix with first row [2 2 1], second row [2 1 1], and third row [1 1 2].
(5 pts each; requires justification) (a) Is this graph simple? (b) Is this a tree? (c) Is it reducible?
(20 pts) Write a matrix representing how many ways you can travel from each node to each node with three edges.
(15 pts) Draw three distinct homeomorphically irreducible trees of order eight.
Discrete Math
(5 pts) Write a boolean expression involving variables x, y which evaluates to true when any one of them is true.
(10 pts) Write a boolean expression involving variables x, y, z which evaluates to true when all three are false.
(20 pts) Write a boolean expression involving variables w, x, y, z which evaluates to true when all three are false and false when all three are true.
Proofs
(5 pts) Show an argument for the Pythagorean theorem.
(5 pts) Write a false proof and explain why it fails.
(15 pts) Prove using induction that 1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2.
(20 pts) Prove using the method of infinite descent that the square root of two is irrational.
Math in Biology
(5 pts each) Draw a planar tesselation using (a) equilateral traingles, (b) squares, and (c) hexagons.
(15 pts) Write a proof for why no other regular polygon can tesselate the plane.
Hands on Activities
Graph Theory
(10 pts) Recreate the Konigsburg bridge problem using origami paper and string.
(15 pts) Add two bridges to Konigsburg to make it solvable.
Topology
(5 pts) Create a mobius strip.
(5 pts each) Find an object with (a) positive Guassian curvature, (b) negative Gaussian curvature, and (c) no Gaussian curvature.
(5 pts) Find two objects that are topologically equivalent to each other.
Complex Numbers
(5 pts) Use string to represent the axes of the complex plane.
(10 pts) Express 4 + 3i using string on that axes.
(10 pts) Demonstrate a multiplication by e^(2*pi*i/3) by rotating string about the origin.
Math in Nature
(5 pts) Using string, make a catenary.
(10 pts) Approximate a Gaussian distribution using string.
(15 pts) Using three straws and hot glue, create a tetrahedron.
(5 pts) Dip the tetrahedron in bubble mix and see resulting shape.
Math in Music
(5 pts) Using slinkies, create a standing wave.
(15 pts) Assuing every straw's base pitch is 440 Hz, cut a straw to play at around 1320 Hz.
Math in Physics
(5 pts) Use string to create a closed orbit of a planet around another planet.
(15 pts) Use your closed orbit and origami paper to demonstrate Kepler's second law.
(5 pts) Approximate the curve that solves the brachistochrone problem using paper or string.
(10 pts) Use a technique to draw exactly the above curve.