8 : Elizabeth : Discussed polite numbers i.e. integers that can be written as a sum of
two or more consecutive positive integers.
9 & 10 : Srinivasa : Number of edges in a complete graph K3, K4, K5.
Proved that #edges(K_n) = n(n-1)/2 by induction.
Counted the number of two-colourings of K3, then listed
them by writing down a binary counting table from 0 to 7.
Showed any 2-colouring of K6 has a mono-colour triangle.
Showed 2-colourings of K5 need not have a mono-colour triangle.
11 & 12 : Ralph : Computation of the perimeter and area of the Snowflake curve.
8 & 9 : Tamiru : Modular Arithmetic.
10: Srinivasa : Proof of infinitude of primes.
Program to exchange 2 variables with a temporary variable, then without.
Counted the number of moves to solve Towers of Hanoi problem. This
introduced them to recursion --- starting from the simplest cases of 1 and 2 disks.
11&12 : Ralph : Moduli spaces.
(1) The parametrisation of a semi-cubical parabola is a line.
(2) The parametrisation of right-angled triangles is a curve in R^3.
(3) The parametrisation of all affine lines is an open Moebius band.
8 & 9 : Ralph : Computed sum of consecutive squares from 1 up to 24 (stacking cannonballs).
Discussed the cannonball vector and its crazy "length".
Competed reciting digits of pi and powers of two.
Revealed the Newton method for computing a square root.
10: Srinivasa : Induction --- proved the sum of the first n natural numbers,
the sum of the first n squares, and the minimal moves for Tower of Hanoi.
Showed that (a^2+c^2)/2 + b^2 > ab + bc
Q: What is the relationship between GCD and LCM of a and b? (UNFINISHED)
11& 12 Peter : How many ways can you tie a tie (with n moves -- left (L), right (R) or up (U)) ?
8 & 9 : Elizabeth : Graph theory. Started with Bridges of Koenigsberg problem.
Discussed Eulerian paths and circuits, as well as the theorem
for when graphs do or do not have Eulerian paths or circuits.
10: Zoltan : Horrible Game - Erauqs.
Josephus Problem --- tabulated values.
AMC Inter 2013.
11& 12 : Ralph : We examined the volume and medians of a golden tetrahedron.
We also ran sanity checks on the equations ... and even constructed models.
(A,B,C,D,E,F) = (63,57,54,41,35,38).
8 & 9 : Ralph : Reminded the students about N, Z, Q, R, C.
Also exposed them to existence of quaternions, octonions, sedenions, ...
Played with continued fractions and computed them for sqrt(d) for
squarefree d between 2 and 17.
10: Zoltan : Horrible Game - sum_{digits of x} 2^digit
Josephus Problem --- remains homework.
Continued AMC Inter 2013 question.
11& 12 : Peter : Geometric variational Problems.
First variation formula for length (acceleration = 0 --> straight)
(geodesic equation).
First variation for area (mean curvature H = 0, i.e. ave of curvature = 0).
8 & 9 : Elizabeth : Small math problem to break the ice.
Reminder of graph theory. Planar graphs, complete graphs and
complete bipartite graphs. Discussed which are planar ... wrote down theorem.
Asked what is planar on a torus?
10: Ralph : Reminded the students about N, Z, Q, R, C.
Also exposed them to existence of quaternions, octonions, sedenions, ...
Played with continued fractions and computed them for sqrt(d) for
squarefree d between 2 and 17.
11 : Srinivasa : Sum of first n natural numbers.
Tower of Hanoi problems.
12 : Peter : Calculus, differential equations, acceleration is constant --> gravity.
motion in a circle. Two dimensional gradients, higher dimensional
and steepest descent.
8 : Ralph : 3-Powerful numbers. Discussed how to bound and find them.
Moebius band and Klein bottle. Proved MB + MB = KB.
9 : Tamiru : Proof of Pythagoras' theorem.
Mid point theorem.
Given a rectangle ABCD and any interior point P
proved that (AP)^2 + (CP)^2 = (BP)^2 + (DP)^2.
10: Zoltan : 2013 AMC questions solved on board.
one as a dramatic theatre event.
11 : Srinivasa : Size of the powerset of a set by induction.
Boolean algebra : De Morgan's laws and relevance to programming.
Towers of Hanoi variants.
12 : Peter : Cartesian plane, coordinate rotations, matrices, preserve inner product
even and odd part of f(x), f(x)*f(-x) = f(^+)^2 - f(^-)^2
"imaginary angle" rotations,
cosh(x), sinh(x) preserve Minkowski metric.
Lorentz transformations.