8 : Anand : Some elementary computations to begin with.
This was followed by Cantor's diagonalisation to show
there is no bijection between N and R (in simpler words).
9 : Chris : The ‘clock game’ – using digits (419) to make targets from 0 to 51.
Exploring factorials, fractional roots and other strange notations.
10: Tamiru : a) Diagonal of a unit regular pentagon is golden ratio.
b) Use induction to prove 2^n > n^2 for n >=5.
11+12:Ralph: N < Z < Q < R < C < H.
Compared the effect of i on the complex plane to
the effect of the 2x2 matrix J = [[0, -1], [1, 0]]
on the Cartesian plane.
Both are rotation by 90 degrees anticlockwise.
Gave the isomorphism from a+ib --> [[a, -b], [b, a]].
Finished with the quaternions H={a+ib+cj+dk}.
8 : Tryon : Mathemagic.
9&10 : Chris : Angle sum of points in an n-pointed star
including "turning number" t.
11+12:Ralph: Solving equations without solving equations.
Positivity : 2x+4 = 0 in N.
Divisibility : 3x+5 = 0 in Z.
Squares : 9x^2-4 = 0 in Q.
Contradiction : x^2-2 = 0 in Q.
Positivity : x^2+172y^6+42z^42 + 5 = 0 in R.
Legendre symbol for solving x^2 = a (mod p).