2021 Term 2
Term 2 --- Week 1
8 : Ralph : Elementary Topology. The disk, cylinder, sphere, Moebius band,
torus, Klein bottle, and real projective plane. Identification of
edges of a square models. MB + MB = KB
9 : Zoltan : Some problems from MCYA Noether 2002.
10 : Peter : Math structures I, numbers we're familiar with and inventing new ones.
N, Z, Q, R, C = R(i), Q(sqrt{2}), ...
Division of complex numbers.
11 : David J. : Addition chains and strategies for finding them.
12 : Tamiru : Numerical semi-group and examples.
Started proving the Chicken McNugget Theorem.
Term 2 --- Week 2
8 : Elizabeth : Introductory graph theory. Eulerian paths and Hamiltonian circuits.
9 : Tamiru : Number system. Rational numbers (repeating and terminating decimals).
10 : Chris : Introduction to metrics and distance functions. We then tried putting
metrics on different sets.
11+12: Michael : Construction of natural numbers as equivalence classes of finite sets.
Cardinalities of N, Q, R. Does there exist w : |N|< w < |R|?
Diophantine puzzles. Solve
(a+b=ab), (a+b=cd, c+d=ab), (a+b=cd, c+d=ef, e+f=ab), etc ...
Term 2 --- Week 3
8 : Celina + : Introductory predicate calculus.
Ralph Negation, or, and, De Morgans Law, truth tables.
Proof that sqrt{2} is not rational.
9 : Elizabeth : Introductory graph theory. Eulerian paths and Hamiltonian circuits.
Euler's theorem.
10 : Peter : Special relativity via cosh and sinh (c.f. sin and cos for 2-D rotations).
11 : Chris : Contraction mappings and generalised distances.
12 : Tamiru : Numerical semigroup problems. Proofs of theorems.
Term 2 --- Week 4
8 : Ralph : Computed Euler characteristic of Platonic solids and various other
polyhedra. Proved Euler's Theorem for planar graphs and applied
to Platonic solids and other polyhedra.
9 : Zoltan : Continued with problems from Noether 2002.
Student solved P9 on board --- speeds and times.
10 : Tryon : Locker Up, Tryon's Troublesome Tribulations, math and logic puzzles.
11+12: Peter : Special relativity via cosh and sinh (c.f. sin and cos for 2-D rotations).
Term 2 --- Week 5
8 : Tamiru : Base number systems.
9 : Zoltan : Continued with problems from Noether 2002.
Student solved Q13 and Q7 on board.
10 : Tryon : Locker Up, Tryon's Troublesome Tribulations, math and logic puzzles.
Finished with probabilistic card tricks.
11+12: David : Game theory.
Term 2 --- Week 6
8 : Tamiru : Base number systems, binary, hexadecimal, octal
and other base conversions.
9 : Ralph : Computed Euler characteristic of Platonic solids and various other prisms and
polyhedra. Proved Euler's Theorem for planar graphs and applied it
to Platonic solids and other polyhedra.
10 : David : Game Theory, NIM and the usual suspects.
11 : Peter : Fibonacci sequence, general linear recursive sequences,
matrix representations, eigenvalues and eigenvectors.
12 : Zoltan : "101 Problems in Algebra" AMT Book 18.
Solved problems 1,2,3 at least. Mostly individual work.
Term 2 --- Week 7
8 : Peter : Pascal's triangle and algebra. Place notation, bases,
long multiplication, long division in Z and Z[x].
"n choose i" = n! / (i!(n-i)!)
(x-1)(x^n+...+x+1) = x^{n+1}-1
9 : Ralph : Circles, Z points, Q points, R points, C points.
Pythagorean triples. Algorithms to go from Q --> R and R-->Q.
Proved 0.49999... = 0.5, prove Harmonic sum diverges.
10 : Elizabeth : Graph theory - intro definitions then Euler's theorem V-E+F=2.
Hamiltonian circuits ... plus other fun graph theoretic facts.
11+12: Zoltan : Continued with Probs 1-15 of "101 Problems in Algebra"
AMT Book 18. Solved problem 8 on the board.