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.
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.
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.
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.
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.
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.
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.