8 : Zoltan : 2003 ToT Avanced Q1.
Proof infinitely many primes.
2003 ToT Ordinary Geometry.
9 : Angus : Played Order versus Chaos.
10 : Peter : Math structures N, N\{0}, Z, Q, R.
Meaning of fractions and their decimal representations
How to recover a/b from 0.d1d2d3d4... and find the
maximal length k of 1/p as a decimal.
Theorem : Length k | p-1.
11&12: Ralph : Solving e^x = (x-2)^2 via various methods.
Graphing, binary search, linear iteration, Newton's method.
8 : Zoltan : Bridges of Koenigsberg.
Functional equations e.g. f(x+y)=f(x)+f(y)-3.
Horrible Game y = 400-x^2
9 : Ian : Monty Hall Problem
Gathered stats by playing in pairs, 10 games with stay strategy
10 games with switch strategy, added counts, computed observed probs.
Probability theory - prob of union, conditional prob.
Derived prob of each strategy winning.
Generalised to n doors.
10 : Peter : Proved the theorem that length k of recurrence of 1/p satisfies k | p-1.
Continued playing with related questions on fractions.
Also worked on Z and C.
11&12: Thalia : Geodesics, Black holes, singularities, differential equations,
evolution of surfaces.
8 : Zoltan : Almost calculated expectation(X1-X2) where Xi in Uniform(0,1).
Excessive diversions e.g. Graham's number.
9 : Angus : Played the Wombat game (though many opted for Hangman).
10 : Ian : Monty Hall Problem & generalisation to n doors where
host shows 1 door versus host shows n-2 doors.
11&12 : Ralph : Discussed definition of derivative which led to the
difference between arithmetic, algebraic and differential equations.
Discussed singularities of equations and relation to tangent space.
8&9: Zoltan : Watch Numberphile on Vampire Numbers.
How many squares, rectangles in an nxn grid.
Looks like 1^3+...+n^3=(n(n+1)/2)^2.
Also counted the number of triangles in a triangular grid.
10 : Angus : Lengths on a lattice.
Side conversations on calculus.
11&12 : Ralph : Discussion of continuous vs discrete.
Anomaly :- High energy cosmic ray anomaly,
Bound :- Bekenstein Bound,
Catastrophe :- Ultraviolet catastrophe,
Dilemma :- Banach-Tarski paradox,
Experiment :- Stern-Gerlach experiment.
8 : Peter : Pascal's Triangle, (n-th row =11^n in any base).
In base p have 11^p=(p+1)n=...
Similarly, (p+q)^n=p^n+(n//1)p^{n-1}q+...
Proved binomial Theorem. Established why (n//i) is called "n choose i".
Proved binomial formula and mentioned non-commutative extension.
9 : Zoltan : AMC question, Square with vertices ABCD and point P inside.
Find area of triangle ABP if AP=51, DP=53, CP=25 subject
to the constraint that and all other lengths are integers.
Pythagorean triple formula, checked it and stated it gives all.
AM-GM inequality. Rambled about 1729.
Horrible game y=x+max prime factor(x).
10 : Ian : More on lattices, sums of squares.
11&12 : Ralph : p-adic numbers. Definitions of Z_p and Q_p.
Showed N, -1, Z all in Z_5. Stated Q is in Q_5.
Described the p-adic length.
Equation solving in 5-adics and found a square root of -1 in Q_5.
8 : Peter : Proved some patterns in Pascal's Triangle (e.g. #(n//i)=0 mod k)
Pascal's tetrahedron and (p+q+r)^n.
Binomial and multinomial distributions.
prod{i=1}^n (p_i+q_i)
prod{i=1}^n (x+a_i)
9 : Ralph : Matrix Rings (e.g. M_{2x2}(Z)).
What is a matrix, what is A+B?, what is AxB?
Identity matrix, x is non-commutative, + is associative, ...
Solving matrix equations like A^2 = -I.
10 : Angus : Regular tilings of R^2.
Can you tile a shpere?
11 : Ian : Monty Hall experiments, probability theory
Bayes Theorem, conditional probability, COVID tests.
12 : Zoltan : AMC 2013 Senior problems.
8&9: Colin, : Origami and mathematics. Build the Octahedral ball.
Elizabeth
&Ian
10 : Thalia : Detailed setup for finite state automata.
&Peter Described matrix solutions to simultaneous equations.
Vector spaces, bases, linear operators, column vectors and matrices.
11&12 : Angus : Tilings of the plane, tilings of the sphere.
& Zoltan