8 : Tryon : Infinity
9 : Peter : Sum of angles in a triangle is 180 degrees.
There are an infinite number of primes.
Countable sets, N, Z, Q.
Q is countable but R is uncountable.
In the decimal expansion of 1/n = 0.d1 d2 ... dk then k <= n-1
and in fact k divides n-1.
10 : Zoltan : AIC (CAT) 2011 and 2013 problems.
11&12 : Srinivasa : Public key cryptography --- RSA --- maths.
8 : Tryon : Puzzles
9 : Peter : (A) Length, l(p), of repeating decimal for 1/p
(1) l(p) <= p-1 using remainders,
(2) l(p) divides p-1 using Fermat's little theorem.
(B) proofs by induction
10 : Zoltan : AIC (CAT) 2011 and 2013 problems continued.
11&12 : Paul : Irrationality.
Proved that sqrt(2) and log_2(3) are not in Q.
If a1,a2 in Q and b1,b2 not in Q then
-- a1 + a2 in Q
-- a1 + b1 not in Q
-- b1 + b2 in ?
-- a1*a2 in Q
-- a1*b1 not in Q
-- b1*b2 in ?
Defined algebraic numbers and showed i, sqrt(2) are algebraic.
Defined transcendental numbers.
Showed that one of (pi+e), (pi*e) is irrational
given that both pi and e are transcendental.
8 : Peter : Pascal's triangle, proofs by induction
(i) sum of row n in Pascal's triangle = 2^n
(ii) sum of first n natural numbers = n(n+1)/2
(iii) sum of squares of first n natural numbers = n(n+1)(2n+1)/6
(iv) sum of cubes of first n natural numbers = [n(n+1)/2]^2
9 : Zoltan : Horrible game y=(x-3)(x+1)
Median of a triangle bisects the area, and lines parallel
to the base (but not the angle).
10 : Paul : Irrationality.
Proved that sqrt(2) and log_2(3) are not in Q.
If a1,a2 in Q and b1,b2 not in Q then
-- a1 + a2 in Q
-- a1 + b1 not in Q
-- b1 + b2 in ?
-- a1*a2 in Q
-- a1*b1 not in Q
-- b1*b2 in ?
Defined algebraic numbers and showed i, sqrt(2) are algebraic.
Defined transcendental numbers.
Showed that one of (pi+e), (pi*e) is irrational
given that both pi and e are transcendental.
11&12 : Srinivasa : computational complexity of addition and multiplication
(i) program to swap values in two integer variables
using a temporary variable or otherwise.
(ii) intuition behind computational complexity
of bit addition, multiplication leading to
Karatsuba.
8 : Peter : Pascal's triangle continued.
(p+q)^n = ...
Binomial Theorem
9 : Zoltan : Horrible game y=x^3+1000
Four triangle centres.
Proof of Thales Theorem
10 : Paul : Geometry. Define isometry and reflection.
Prove a reflection is an isometry.
Reflections in two parallel lines is a translation.
Reflections in two non-parallel lines is a rotation.
Show the three reflection theorem.
11&12 : Tryon : Fractals
8 : Peter : Pascal's triangle continued.
(n k) := n choose k.
Pascal's tetrahedron. Multinomial coefficients.
Non-commutativity, quantum computing (& magic).
9 : Zoltan : Horrible game y=x^2 mod 19
Prove medians of a triangle trisect each other
and hence are concurrent.
10 : Tryon : Infinity.
11&12 : Paul : Geometric isometries. Three reflections theorem.
8 : Peter : Pascal's triangle and tetrahedron continued.
Distributivity.
Patterns in Pascal's triangle (mod 2, mod 3, ...)
prod(x+a_i), prod(x_i+a_i), etc.
9 : Ralph : Horrible game y=Collatz function. Best quote:
"Now I understand why its called the horrible game!"
The Collatz conjecture about the iterated function.
10 : Tryon : Fractals.
& Elizabeth
11&12 : Paul : Principle of inclusion & exclusion.
8&9 : Elizabeth & Zoltan : Question 1 from "Parabolic Problems" book.
10: Ralph : Horrible game y=Collatz function.
The Collatz conjecture about the iterated function.
11&12 : Srinivasa : RSA cryptology computational aspects.
Addition chains, multiplication via high school algorithm
and Karatsuba algorithm. Divide and conquer.