Clone of Canberra Mathematics Enrichment programme
The Canberra Mathematics Enrichment programme is about enrichment, rather than acceleration, through the school syllabus. Students are exposed to a range of extracurricular material, with an emphasis on solving problems in a fun and relaxed atmosphere. They have the option to participate in the International Tournament of Towns, an international competition organised from Moscow that Canberra students have participated in for many years.
The aims of the programme are to:
 foster enjoyment in mathematics;
 develop problemsolving skills;
 prepare for the Tournament of the Towns competition.
It is organised with the Australian Mathematics Trust at the University of Canberra.
Contact us
We encourage parents, teachers and principals to email maths.enrichment@anu.edu.au with recommendations of capable students whom they think would benefit from the programme.
For all other enquiries about the program email admin.teaching.msi@anu.edu.au.
Sessions
We run two types of sessions. The first and fourth terms of each year are preparation for the Tournament of the Towns exams during which we practise previous exams and learn new problem solving skills. The second and third terms are general enrichment sessions during which the enrichers get to enthuse the students with whatever mathematics they feel like. Here are some past sessions we've run:
2019
Term 2  Enrichment
Week 1
Year  Teacher  Topic 

8 
Peter 
Proofs, Deduction, e.g. AMGM inequality. 

Ellen 
Contradiction, e.g. AMGM inequality Contradiction, e.g. infinite number of primes Deduction, e.g. sum of 1+2+3+...+n=n(n+1)/2 Induction, e.g. the sum 1^2+2^2+3^2+...+n^2=n(n+1)(2n+1)/6 Euclidean geometry proofs starting from axioms. 
9 
William 
Introduction to algorithms, sorting (proof of correctness) Bubble sort, quick sort Complexity. 
10 
David 
Analysing the game of covering rectangles with arbitrary size squares Last person to move loses. 
11 & 12 
Kifan 
Preliminary description of RSA. 

Ralph 
History and creation of public key cryptography RSA encryption and decryption RHB encryption and decryption (multiplication vs exponentiation) Euclid's algorithm to break the latter Euler phi function, Euler's Theorem Magma experiments with RSA and Factorisation. 
Week 2
Year  Teacher  Topic 

8 
Peter 
Observation that sum_{i=1..n} i^3 = T(n)^2. Proof by induction Proof by deduction summing a standard nxn multiplication table two ways
Note that each gnomon divided by i is the sum of NWSE diagonals of ixi squares filled with 1s ... hence i^2 Proof that n^2 = sum of 1st n odd numbers by picture. Doing algebra mentally: (k+1)^4 (16 product terms to be grouped, Pascal triangle) Inductive "proof" that in all groups of n people everyone has the same gender Elementary Euclidean geometry proofs: parallelograms exist. 
9 
William 
Colouring, parity, modular and numerical invariants Problems 1,7,10,11,12,14 from invariants worksheet. 
10 
David 
Analysing the game of covering rectangles with arbitrary size squares Last person to move loses. 
11 
Liam 
Towers of Hanoi patterns The traditional variant and a new sliding/rotating version. 
12 
Ralph 
History of factoring algorithms Lookup table, trial division, mod 6 residues, Fermat method Eratosthenes sieve. Comparison of sieve methods using CEP. 
Week 3
Year  Teacher  Topic 

8 
Ellen 
Worked on introductory problems involving invariants For example, can one tile a chessboard with two opposite corners missing with 31 dominoes. 
9 
Peter 
Proofs, Deduction, e.g. AMGM inequality Contradiction, e.g. AMGM inequality Contradiction, e.g. infinite number of primes Deduction, e.g. sum of 1+2+3+...+n=n(n+1)/2 Induction, e.g. the sum 1^2+2^2+3^2+...+n^2=n(n+1)(2n+1)/6 Euclidean geometry proofs starting from axioms. 
10 
William 
Quicksort example exploring complexity Binary search complexity (depends on quicksort) Minimum spanning tree algorithms (Prim, Kruskal). 
11 
Kifan 
Cyclic groups focusing on (Z/p)*. 

Zoltán 
Discrete logarithm problem DiffieHellman key exchange and encryption scheme with security analysis Maninthemiddle attack, key reuse, authentication Comparison to RSA. Brief description of elliptic curves. 
12 
Liam 
Towers of Hanoi and other 2^n puzzles. 

David 
The traditional variant and a new sliding/rotating version. 
Week 4
Year  Teacher  Topic 

8 
Tamiru 
Proof that congruence is an equivalence relation Solution to linear modular congruences. 
9 
Peter 
Observation that sum_{i=1..n} i^3 = T(n)^2. Proof by induction Proof by deduction summing a standard nxn multiplication table two ways.
Note that each gnomon divided by i is the sum of NWSE diagonals of ixi squares filled with 1s ... hence i^2. Proof that n^2 = sum of 1st n odd numbers by picture Doing algebra mentally: (k+1)^4 (16 product terms to be grouped, Pascal triangle) Inductive "proof" that in all groups of n people everyone has the same gender Elementary Euclidean geometry proofs axiom based Axioms of a group, proof that identity and inverses are unique by contradiction. 
10 
William 
Encoding  detecting errors, (parity, modulo prime, mention LFSR) Correcting errors  majority vote, Hadamard codes Measures of encodings  size of codewords, "extra information" detect/correctability. 
11 
Zoltán 
Redux of DiffieHellman key exchange, GCDs, Dlogs. 
12 
David 
Goedel's Theorem 
Week 5
Year  Teacher  Topic 

8 
Tamiru 
If a=c (mod m) and b=d (mod m) then a+b = c+d (mod m) ab = cd (mod m) Solution of linear congruence followed by example of Chinese Remainder Theorem. 
9 
David & William 
Discussion of Goedel's Theorem. 
10 
Zoltán 
1992 Telecom AMOC Junior Contest. 
11 
Ellen 
Colouring, parity, modular and numerical invariants Problems 1,5,10,11,13 from the invariants sheet as a class. 
12 
Peter 
Calculus. Fundamental Theorem Exponential function and its power series Solution to simple 2nd order differential equation Meaning of exp(x), definition of pi by e^(i+pi)+1=0. 
Week 6
Year  Teacher  Topic 

8 
Ralph 
Worked of problems 1, 2 and 4 of Structure of the Reals as a class Discussed N, Z, Q, algebraics, transcendentals, Cantor and Liouville's number L=1/2^0! + 1/2^1! + 1/2^2! + ... 
9 
Peter 
Axioms and examples of groups, rings and fields When is Z/NZ a field Matrices as an example of a noncommutative ring. 
10 
Zoltán 
1992 Telecom AMOC Junior Contest. 
11 & 12 
William 
Turing machines Computability/ Hilbert's problems, definition, state machines, clearing, busy beavers, numerical operations, addition, unary, binary, multiplication, Universal Turing machines, halting problem and solution to Hilbert's 10th problem. 
Term 1  ToT preparation
Week 1
Year  Teacher  Topic 

8 
Ralph 
Why mathematicians do proofs Counting lines & regions created by lines between all pairs of vertices on a circle? Why are triangular numbers triangular Gauss's proof of the formula for the sum of the first n natural numbers (Two minute description of modular forms.) 
9 
Zoltán 
2014JO40spr, Q1 with fully presented proof. 
10 
William 
2014JO40spr, Q1 with fully presented proof Q2 worked out ideas only  unfinished. 
11 
David 
2014SO40spr, Q1. 
12 
Peter 
2014SO40spr, Q1, Q3 complete Q4 part done. 
Week 2
Year  Teacher  Topic 

8 
Ralph 
Sets, N,Z,Q,R,C. Comparisons of sets. Demonstration that N = 2N = Z = Q Cantor's proof that N < R What are complex numbers. 
9 
Zoltán 
2014JO40spr, Q2 with fully presented proof. 
10 
William 
2014JO40spr, Q2, Q3 with fully presented proof. 
11 
Tamiru 
2014SO40spr, Q2. 
12 
Peter 
2014SO40spr, Q4 completed correctly. Q2 started Complex numbers, e^(i*theta), e^(i*pi)+1=0, Fundamental theorem of Algebra, Q,R, C=R(i) Algebraic numbers, Q(sqrt(2)), Q(cuberoot(2)), transcendental numbers N as unary, N in place notation, N in prime power notation. 
Week 3
Year  Teacher  Topic 

8 
Zoltán 
Brussell sprouts (game), 2014JO40spr, Q1 complete, Q2 partially done, The horrible game. 
9 
William 
2014JO40spr, Q3 done, Q4 partial. 
10 
David 
2014JO40spr, Q4, Q5 done. 
11 
Peter 
2014SO40spr, Q3, Q4 completed. Q5 started. 
12 
Tamiru 
2014SO40spr, Q2 completed. Q5 started. 
Week 4
Year  Teacher  Topic 

8 
Zoltán 
Horrible game f(n) = sum_{pprime, pn} p (very very difficult) 2014JO40spr, Q1 Two distinct solutions. 
9 
Ralph 
2014JA40spr, Played with all the questions ... none completed. 
10 
David 
2014JA40spr, Q1, Q2 done. Q3 started. 
11 
Ralph & Tamiru 
2014SA40spr, Q1 2014SO40spr, Q5. 
12 
Tamiru 
2014SO40spr, Q5. 
Week 5
Year  Teacher  Topic 

8 
Kifan 
2014JO40spr, Q2 completed. Q3 started. 
9 
Zoltán 
2014JA40spr, Played with all the questions. Q6 almost complete. 
10 
David 
2014JA40spr, Q3, Q4 almost complete. 
11 
Ralph 
2014SA40spr, Q2 started. Completed arbitrary triangles. 
12 

Week 6
Year  Teacher  Topic 

8 
Peter 
2014JO40spr, Q4 and Q5 partially done. Introduction to induction. 
9 
Liam & Chris 
2014JA40spr, 2014JA40spr, Q9 complete. 
10 
David 
2014JA40spr, Q3, Q4 completed. 
11 
Ralph 
2014SA40spr, Q2 completed proof. Required following: Discussed continuous and discontinuous functions Discussed the intermediate value theorem. 
12 
Zoltán 
2014SA40spr, Q5 partial progress, found 4 < best <= 8. Found a winning strategy with 8 moves for Peter. Proved that there is none with 4 moves for Peter. 
Week 7
Year  Teacher  Topic 

8 
Zoltán 
2014JO40spr, Q4 wrote up proof Horrible game  Triangular numbers 2014JO40spr, Q5 Ali Baba repeated n=2,3,4,5, failed n=6 Challenge 1^3+2^3+3^3+...+n^3. 
9 
Liam 
2014JA40spr, completed Q3. 
10 
Ralph 
EFM. 
11 
David 
2014SA40spr, Q3. 
12 
Kifan & Tryon 
2014SA40spr, Q5. 
2018
 2018 Term 4: ToT preparation with JO2013F, SO2013F, 2013_Fall_JA, 2013_Fall_SA
 2018 Term 3: Enrichment
 2018 Term 2: Enrichment
 2018 Term 1: ToT preparation with JO39aut, JA39aut, SO39aut, SA39aut
2017
 2017 Term 4: ToT preparation with JO38spr, JA38spr, SO38spr, SA38spr
 2017 Term 3: Enrichment
 2017 Term 2: Enrichment
 2017 Term 1: ToT preparation with JO38aut, JA38aut, SO38aut, SA38aut
Resources
The Australian Maths Trust (AMT) produces the Mathematics Challenge for Young Australians booklets. They form an excellent staged mathematics enrichment program, and are available for various year groups:
 Ramanujan (4+5)
 Newton (5+6)
 Dirichlet (6+7)
 Euler (7+8)
 Gauss (9)
 Noether (9+10)
 Pólya (10)
You can get the booklets directly from AMT (is this the correct link?) or you can ask your school. Each category comes as three separate volumes, "Student Notes" which contain background theory and examples, "Student Problems" and "Teacher Reference Notes".
Enricher material
Teacher  Resource 

Allan 

David  
Ellen  
Fiona  
Peter 

ραλφ  
Tamiru  
Tryon 

Zoltán 

Canberra
 Alfred Deakin High School (Bácskai)
 Australian Maths Trust
 ANU Extensions
 ANU Maths Day
 Canberra enrichment opportunities (Carty 2013)
 Canberra High School
 Canberra Mathematics Enrichment Program
 Dickson Mathematics Enrichment
 Hawker College (SheikhBuchholz)
 National Mathematics Summer School
 Yarralumla Primary School  ChaMPS (BácskaiPrice)
Australia
 CSIRO  STEM Professionals in Schools
 Gifted Students Resources
 University of Melbourne (Melbourne, VIC)
 UWA (Perth, WA)
 Swinburne (Melbourne, VIC)
 Tournament of the Minds
 UNSW Mathematics Enrichment Club (Sydney, NSW)
 USQ Mathematics Enrichment (Springfield, QLD)
Web
 Dimensions  Visualise the 4th dimension
 Khan academy
 Magma calculator
 Numberphile
 OnLine Encyclopedia of Integer Sequences
 Primary School Enrichment  UK
 Proof Wiki
 Simon Marais Mathematics Competition
 Three Blue One Brown
 Tournament of the Towns Moscow, Toronto, Wiki
 VSAUCE
Pebbles
The latest bits of mathematics that some of us find interesting, enjoy.