2018 Term 3

Term 3 --- Week 1

8     : Ralph & : Symmetries of a triangle, --> D_3 (dihedral group)
        David     D_3 multiplication table.
9     : Tamiru  : Polynomial arithmetic, rational root theorem, 
                  Problems from Number theory and statistics.
10    : William : Message encoding, error detection/correction,
                  Parity, LFSR/CRC, Hadamard encoding, Hamming distance,
                  Viterbi encoding.
11    : Zoltan  : MCYA Polya 1993 questions..
12    : Peter   : RSA encryption, Fermat's factoring method, Number field sieve.

Term 3 --- Week 2

8     : Ralph   : Four 4s problem (+,-,x,/,parentheses,powers) from 1 .. 20
                  Also allow square roots and factorials to get the missing ones.
                  Express as sum of 3 palindromes (a) 38751, (b) 98512, (c) 123456789.
9     : Tamiru  : Rational roots of an integer polynomial followed by
                  Number theory and statistics problems.
10    : William : Modular arithmetic, addition, multiplication, powers.
                  Pascal's triangle and polynomial expansion.
                  Problems from The Sock Drawer.
11    : David   : Symmetries of an equilateral triangle, D_3,
                  Rigid symmetries of a tetrahedron, S_4.
                  This is the same as the relabellings of a tetrahedron!
12    : Peter   : Signatures, cryptocurrency and blockchains.

Term 3 --- Week 3

8     :Elizabeth: Questions 2, 4 and 6 from this NRICH paper
       & Ben      Twin prime discussions motivated by Q4.
9     : David   : Squaring the circle, duplicate the cube, trisect the angle.
                  Permutations their composition and parity and application:
                  Rubik's cube can only be solved with even permutations.
                  So one cannot flip a single middle edge piece!
10    : Peter   : Groups, Rings and Fields. (In preparation for RSA)
11    : William : Entropy, expected values, run-length encoding, Huffman trees,
                  Prefix-free-ness (with reference to Morse code).
12    : Ralph   : Find 1 non-zero integer solution to each of the following equations.
                  (a) x^2+y^2=z^2, (b) x^2+y^2=2z^2, (c) x^2+y^2=3z^2. 
                  How about 1 non-zero solution in Z/3Z?
                  Four 4s problem. Three palindromes problem.

Term 3 --- Week 4

8     : Asilata : Young diagrams, Young tableaux, Robinson-Schensted correspondence
       & Ben      
9     : Ralph   : Four 4s problem from 1 to 20, first without sqrt, factorial---then with.
                  Three palindromes problem ... 3 examples.
                  Add two Moebius bands to create a Klein bottle.
10    : Peter   : Decsiption of RSA, euclidean algorithm for GCD of two integers.
11    : William : Detecting errors, correcting errors, information rate, majority vote,
                  parity, modulo prime, hash functions (CRCs), hamming codes,
                  hadamard codes, viterbi encoding.
12    : David   : Proofs of impossibility.
                  Squaring the circle, duplicating the cube, trisecting the angle.
                  Rubik's cube can only be solved with even permutations ---
                    so one cannot flip a single middle edge piece!
                  Tiles in the 15-14 puzzle cannot be reversed.

Term 3 --- Week 5

8     : David   : Proofs of impossibility.
                  Squaring the circle, duplicating the cube, trisecting the angle.
                  Rubik's cube can only be solved with even permutations ---
                    so one cannot flip a single middle edge piece!
                  Tiles in the 15-14 puzzle cannot be reversed.
9     : Ralph   : Find non-zero integer solutions to x^2+y^2=n*z^2, for n=1,2,3.
                  Use mod 3 argument to show the last one has no solutions.
                  Find all solutions of n=1 case by chord method.
10    : Peter   : Structure of Z/NZ when N=p*q.
                  GCD algorithm in Z and Z[x].
                  Factorisation of N via solutions to x^2=y^2 (mod N).
                  Reminder that GF(p^n) := GF(p)[x]/<f(x)>.
11+12 : William : Compression - entropy, expected value/cost, run length encoding
                  Huffman encoading, statistics for 1-grams to 9-grams in english.

Term 3 --- Week 6

8     : Tryon   : Problems from The Sock Drawer.
9     : Tamiru  : Problem 1 and 2 from Number Theory and Statistics.
10    : David   : Proofs of impossibility.
                  Squaring the circle, duplicating the cube, trisecting the angle.
                  Rubik's cube can only be solved with even permutations ---
                    so one cannot flip a single middle edge piece!
                  Tiles in the 15-14 puzzle cannot be reversed.
11    : Ralph   : Four 4s problem - express 1..20 using (+,-,x,/,parentheses,powers).
                  When they fail to express 11,13 allow sqrt and factorial.
                  Topology - proof that sum of two Moebius bands is a Klein bottle.
12    : Ben     : Knot theory - link invariants, Reidemeister moves, 
                  basic +/- link invriant, constructing the Jones polynomial
                  via the Kaufmann bracket.

Term 3 --- Week 7

8     : David   : Solutions to a quadratic, cubic and quartic.
                  Permutations and applications to non-solution of quintic.
9     : Tamiru  : ?.
10    : Ralph   : Four 4s problem - express 1..20 using (+,-,x,/,parentheses,powers).
                  When they fail to express 11,13 allow sqrt and factorial.
                  Topology - proof that sum of two Moebius bands is a Klein bottle.
11    : Peter   : Rings, R x S, Z/NZ = Z/pZ x Z/qZ (when N=pq), RSA.
12    : William : Encoding and error detection and correction.
                  Parity, majority vote, hashing (mod P, LFSR, CRC), hamming code.
                  Briefly talked about Viterbi algorithm.