2023 Term 3
Term 3 --- Week 1
8 : Ralph : Discussions of N, Q, R, C
Transcendentals versus algebraic numbers.
Proof of the quadratic formula.
9 : Angus : PROMYS Logic puzzles then had a teams tournament with critiquing.
10 : Zoltan : 2013 AMC Intermediate problems
Horrible game y = greatest prime factor(x)
11&12: Peter : Review groups, rings, fields with examples.
Vector spaces.
Term 3 --- Week 2
8 :Elizabeth: Introduction to graph theory.
Planar graphs, complete graphs, complete bipartite graphs.
9 : Angus : PROMYS Logic puzzles then had a teams tournament with critiquing.
10 : Zoltan : 2013 AMC Intermediate problems
Horrible game y = x in base 4
11&12: Peter : Vector space over field K (e.g. K^n), module over rings (e.g. <nZ>).
Number fields, ring of integers.
Non-unique factorisation of integers versus unique factorisation
of ideals.
Term 3 --- Week 3
8 : Ralph : Autopowerful numbers, e.g. 3^3+4^4+3^3+5^5 = 3435.
Powerful numbers (e.g. 1^3+5^3+3^3=153). Bound number of solutions.
Pseudopoweful numbers.
9 :Elizabeth: Introduction to graph theory.
Planar graphs, complete graphs, complete bipartite graphs.
10 : Zoltan : Continued fractions, e.g. a_1=[1;1,1,1,1,1,...] = ?
a_2 = ?
Sum of 1/2^n and sum of 2^n.
Horrible game y=(x-4)^2+4
11&12: Peter : Vector spaces, basis, dimension
Rotation of standard R^2 basis as 2x2 matrices in SO(2).
e^x = sum(x^k/k!).
even(e^ix)=:cos(x), odd(e^ix)/i=:sin(x) --> cos^2(x) + sin^2(x) = 1
even(e^x)=:cosh(x), odd(e^x)=:sinh(x) --> cosh^2(x) - sinh^2(x) = 1
Lorentz transformation. Fibonacci vector space.
Term 3 --- Week 4
8&9: Zoltan : Horrible game (1) y=x(20-x)+1, (2) y=sigma_1(x)=sum_{d|x}d.
Perfect numbers, Mersenne Primes.
Why does sigma(36) = sigma(4) x sigma(9) ?
10,11,12: Tamiru: Proving formulae with and without induction.
1^2+2^2+...+n^2=n(n+1)(2n+1)/6.
1^3+2^3+...+n^3=n^2(n+1)^2/4.
(1-1/sqrt{2})(1-1/sqrt(3))...(1-1/sqrt{n}) < 2/n^2 for all n >=2.
Term 3 --- Week 5
8&9: Zoltan : Discussion of "Sleeping beauty Paradox" --- Numberphile.
What is this stuff on the board?
Formula for s_n = sum_{k=1}^{n} k^2 = tetrahedral_{n}+tetrahedral_{n-1}
= n(n+1)(2n+1)/6.
10:Elizabeth : Planar graphs, complete graphs, complete bipartite graphs.
Discussion of graphs on a torus.
11,12: Ralph : Count the number of representations of the integers 0..15
as a sum of the squares of exactly three integers, where
sign and order are considered different.
Provide two proofs that the number of solutions is finite.
Study Jacobi's theta function and it's first 4 powers in Magma
and compare the coefficients of the modular forms and the counts.
Term 3 --- Week 6
8 : Peter : N, Z, Q, R, ... , C.
We can count in N, Z and Q. R is uncountable.
Q, R and C are fields.Recurring decimals are in Q.
9 : Zoltan : AM-GM ... motivation, application, proofs.
10 : Angus : Bramagupta Fibonacci identity.
Sum of squares in two ways ... e.g. 7^2+4^2 = 8^2+1^2.
11,12: Ralph : We continued with modular forms.
In particular, we would have discussed SL(2,Z) and
operations in that group and connected it to MFs
and its action on the upper half complex plane.
Term 3 --- Week 7
8 : Peter : Groups, rings, fields, Z/NZ, units group.
Z/NZ is a field if and only if N is prime.
9 :Elizabeth: Discussed polite numbers ... i.e. integers that
can be written as a sum of two or more consecutive integers..
10 : Angus : Paper folding and binary expansions.
11 : Tamiru : ???
12 : Ian : Monty Hall, probability axioms, conditional probability,
Bayes theorem using COVID test examples.
Term 3 --- Week 8
8, 9, 10, 11 & 12 : Vandita : All classes were put in one room and were regaled
by the leading UK mathematician Dr Vandita Patel.
There was amazing paper folding followed by a brief lecture.
See her website for more exciting mathematics that she does.