# 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 extra-curricular 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 problem-solving skills;
• prepare for the Tournament of the Towns competition.

It is organised with the Australian Mathematics Trust at the University of Canberra.

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.

## 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. AM-GM inequality.

Ellen

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

1. sum across rows then sum rows to get RHS
2. sum i-th gnomon, (1), (2,4,2), (3,6,9,6,3), ... , to get i^3.

Note that each gnomon divided by i is the sum of NW-SE 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. AM-GM 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

Diffie-Hellman key exchange and encryption scheme with security analysis

Man-in-the-middle 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.

1. sum across rows then sum rows to get RHS
2. sum i-th gnomon, (1), (2,4,2), (3,6,9,6,3), ... , to get i^3.

Note that each gnomon divided by i is the sum of NW-SE 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/correct-ability.

11

Zoltán

Redux of Diffie-Hellman 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 non-commutative 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

2014-JO-40-spr, Q1 with fully presented proof.

10

William

2014-JO-40-spr, Q1 with fully presented proof

Q2 worked out ideas only - unfinished.

11

David

2014-SO-40-spr, Q1.

12

Peter

2014-SO-40-spr, 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

2014-JO-40-spr, Q2 with fully presented proof.

10

William

2014-JO-40-spr, Q2, Q3 with fully presented proof.

11

Tamiru

2014-SO-40-spr, Q2.

12

Peter

2014-SO-40-spr, 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),

2014-JO-40-spr, Q1 complete, Q2 partially done,

The horrible game.

9

William

2014-JO-40-spr, Q3 done, Q4 partial.

10

David

2014-JO-40-spr, Q4, Q5 done.

11

Peter

2014-SO-40-spr, Q3, Q4 completed. Q5 started.

12

Tamiru

2014-SO-40-spr, Q2 completed. Q5 started.

##### Week 4
Year Teacher Topic

8

Zoltán

Horrible game f(n) = sum_{p-prime, p|n} p (very very difficult)

2014-JO-40-spr, Q1 Two distinct solutions.

9

Ralph

2014-JA-40-spr, Played with all the questions ... none completed.

10

David

2014-JA-40-spr, Q1, Q2 done. Q3 started.

11

Ralph & Tamiru

2014-SA-40-spr, Q1

2014-SO-40-spr, Q5.

12

Tamiru

2014-SO-40-spr, Q5.

##### Week 5
Year Teacher Topic

8

Kifan

2014-JO-40-spr, Q2 completed. Q3 started.

9

Zoltán

2014-JA-40-spr, Played with all the questions. Q6 almost complete.

10

David

2014-JA-40-spr, Q3, Q4 almost complete.

11

Ralph

2014-SA-40-spr, Q2 started. Completed arbitrary triangles.

12

2014-SA-40-spr

##### Week 6
Year Teacher Topic

8

Peter

2014-JO-40-spr, Q4 and Q5 partially done. Introduction to induction.

9

Liam & Chris

2014-JA-40-spr, 2014-JA-40-spr, Q9 complete.

10

David

2014-JA-40-spr, Q3, Q4 completed.

11

Ralph

2014-SA-40-spr, Q2 completed proof. Required following:

Discussed continuous and discontinuous functions

Discussed the intermediate value theorem.

12

Zoltán

2014-SA-40-spr, 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

2014-JO-40-spr, Q4 wrote up proof

Horrible game - Triangular numbers

2014-JO-40-spr, Q5 Ali Baba repeated n=2,3,4,5, failed n=6

Challenge 1^3+2^3+3^3+...+n^3.

9

Liam

2014-JA-40-spr, completed Q3.

10

Ralph

EFM.

11

David

2014-SA-40-spr, Q3.

12

Kifan & Tryon

2014-SA-40-spr, Q5.

## 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
• Mathematical Structures I
• Mathematical Structures II
ραλφ
Tamiru
Tryon
Zoltán
• Counting paths on a chessboard

## Pebbles

The latest bits of mathematics that some of us find interesting, enjoy.