8+10 : Zoltan: 2015 AMC paper.
9 : Peter : Pascal's triangle.
11&12 : Paul : Proof that e is irrational.
8+10 : Zoltan : Brussel Sprouts game
Counting rectangles in a grid.
9 : Peter : Pascal's triangle.
Up to multinomial expansions.
Mod k patterns.
p divides p choose j if j is not 0 or p.
distributivity and commutativity.
11&12 : Tryon : Visualising higher dimensional spaces
- flatland & sphereland.
8 : Ralph : Primes, composites and units.
Decomposition of primes as a sum of two squares.
Discovery of the pattern that this is only
possible when p=1 mod 4 or 2 mod 4.
9 : Peter : Pascal's triangle.
product(x+a_i) for i=1..n
Elementary symmetric functions, and non-commutativity.
Quantum computing, bra-ket notation, Shor's algorithm.
10: Zoltan: Proofs of the Theorem of the broken bamboo (also
known as the theorem of Pythagoras).
Brussel Sprouts and Choc Block games.
Remarks on the number 2025 and rectangles.
Horrible game sum of digits squared.
11&12 : Paul : Euler's proof that the sum of the squares of
the inverse of natural numbers is pi squared on 6.
8 :Elizabeth^2: Looked at polite numbers, i.e. positive integers that
can be written as the sum of consecutive positive integers.
9 :Zoltan: Horrible game y=x in base 3.
1/2 = 0.1 in base 2.
Cantor set.
Repeating decimal expressions are rational.
10: Paul: Euler's solution to the Basel problem (Zeta(2)=pi^2/6).
11&12 : Ralph : Knot theory. Simple examples.
Unknot, trefoil, Kauffman bracket, Jones polynomial.
8 :Elizabeth: Graph theory bridges of Koenigsberg.
&Francisco Eulerian paths and circuits.
Theorem of when a graph does or do not have Eulerian paths
or circuits.
9 :Zoltan: Counting rectangles and Niccomachus Theorem.
10: Tryon: Flatland, sphereland and visualising higher dimensions.
11&12 : Ralph : Compute quadratic residues modulo p for 2<=p<=29.
Study Legendre symbol and relate to solubility
of x^2=a (mod p).
Quadratic reciprocity.
8 :Tryon: Flatland, sphereland and visualising higher dimensions.
9 :Zoltan: Horrible game y = P(x) the largest prime factor of x.
Discussion about N subset Z subset Q subset A subset R subset C.
Final comments about 2025 = 1^3+...+9^3=45^2 & rectangles.
10: Elizabeth B: Catalan numbers.
& Peter
11&12 : Ralph : Symmetries of equilateral triangle and regular n-gons.