More about junior sup-maths courses

Course 1-A

Prerequisites

This is the first course in the sequence of our sup-maths courses. Thus it does not require any special knowledge apart from what pupils are taught during Years 1-5, i.e. if you can easily perform basic arithmetic operations then you should not have any problems with understanding the material. However note that some of the problems are quite demanding and many of the pupils will need some time to solve them. Therefore the number one prerequisite is "to be diligent enough" :)

About the content

We will start by looking at some mathematical games in which we will try to find a winning strategy for one of the players. Then there will be several lessons where pupils will be asked to construct some tricky examples. This will show that "even if something looks unrealistic or unachievable, it can still be possible".

Next, in the middle of the course we will talk about logic and there will be a lot of logical puzzles to solve. In particular there will be "knights and knaves puzzles". This part of the course is the hardest, so if you feel very comfortable with such problems, you might want to look at the next courses.

At the end of this course we will have a look at one easy-looking mathematical idea that helps solving a large class of problems. This should show that mathematics is not about ugly formulas but about something much more beautiful that anyone can understand, but not that many people can create.

Next, in the middle of the course we will talk about logic and there will be a lot of logical puzzles to solve. In particular there will be "knights and knaves puzzles". This part of the course is the hardest, so if you feel very comfortable with such problems, you might want to look at the next courses.

At the end of this course we will have a look at one easy-looking mathematical idea that helps solving a large class of problems. This should show that mathematics is not about ugly formulas but about something much more beautiful that anyone can understand, but not that many people can create.

Topics

Course 1-B

Prerequisites

No knowledge of any specialised mathematical theory is required, i.e. easily performing basic operations with numbers is enough to understand the content of this course. However some experience in solving puzzles/olympiad style problems is desirable. In particular if you successfully covered 1-A then you are absolutely welcome (and ready) to join this course :)

About the content

This course is a continuation of course 1-A. Students will discover a few basic mathematical ideas that help with solving a rather big class of problems. And just like in course 1-A there will be a lot of problems where students have to construct examples. In particular, there will a lesson on coming up with algorithms. And as you know, good algorithmic thinking is vital for programming.

Lesson 5 of this course will be dedicated to proofs in general. In particular we will have a look at the examples of "non-proofs" that many students claim to be "full solutions".

The last lesson will be dedicated to problems about selfish but super smart people sharing some sum of money between each other. Some of these problems are hard and one should be extra careful when claiming that "their logic is correct". Actually, when applying to some trading companies, an interviewee can be asked questions similar to these ones...

Lesson 5 of this course will be dedicated to proofs in general. In particular we will have a look at the examples of "non-proofs" that many students claim to be "full solutions".

The last lesson will be dedicated to problems about selfish but super smart people sharing some sum of money between each other. Some of these problems are hard and one should be extra careful when claiming that "their logic is correct". Actually, when applying to some trading companies, an interviewee can be asked questions similar to these ones...

Topics

Course 2-A

Prerequisites

This is the last course where no knowledge of specialised mathematical theory is required, i.e. easily performing basic operations with numbers is enough to understand the content of this course. However, the content of this course strongly relies on course 1-A and just a little bit on 1-B.

About the content

In this course we will mostly talk about some of the key ideas that pop up in the solutions of many olympiad-style problems. E.g. we will dedicate the first lesson to the extension of the idea that "chessboard colouring can help solve a problem" to "some colouring can help solve a problem". Interestingly, this simple observation can be a solution to difficult problems from very serious competitions...

Another key idea is more like a general method to approach some problems which is called "proof by contradiction". Those who have met this before will not face many issues with this topic, whereas those who have not spent enough time proving claims rigorously by contradiction, might find it quite hard. However there is no way we can move on doing more serious mathematics without pupils feeling comfortable with this!

Finally we will spend two lessons on the beginning of combinatorics. We will not mention many definitions (and we will only briefly talk about "n choose k"), but rather talk about the basic things like "product rule", "factorials", "case consideration", ... However note that just like always - there are hard problems that almost do not rely on any theory at all. There will be some in the extra problem sheets at least.

Another key idea is more like a general method to approach some problems which is called "proof by contradiction". Those who have met this before will not face many issues with this topic, whereas those who have not spent enough time proving claims rigorously by contradiction, might find it quite hard. However there is no way we can move on doing more serious mathematics without pupils feeling comfortable with this!

Finally we will spend two lessons on the beginning of combinatorics. We will not mention many definitions (and we will only briefly talk about "n choose k"), but rather talk about the basic things like "product rule", "factorials", "case consideration", ... However note that just like always - there are hard problems that almost do not rely on any theory at all. There will be some in the extra problem sheets at least.

Topics

Course 2-B

Prerequisites

Even though the first topic is variables, it is recommended that students have already met variables before. So this is the number one prerequisite for this course. Also, even though the content does not rely on the previous courses, problem-solving experience is important at this point.

About the content

The first two lessons will be dedicated to variables and problems about them. In particular we will talk about problems where variable(s) play the role of a parameter, e.g problems of the sort "prove that for any group of *n* people such that....". Note that since we expect students to already know what variables are, there will not be many obvious exercises. Instead we will put some interesting problems in the problem sheets.

There will also be one topic about "case consideration". Surprisingly, many people find it hard to find a quick way to go through all the possible cases and this skill is vitally important! So we will work on it during that one lesson.

The next two lessons are about problems that ask students to find the biggest possible/smallest possible amount of something. This type of problems pop up all over the place, and it is sad that many people make silly mistakes and do not know that a correct solution for such problems consists of two parts. We will talk about it and give some quite hard problems to practice.

There will also be one topic about "case consideration". Surprisingly, many people find it hard to find a quick way to go through all the possible cases and this skill is vitally important! So we will work on it during that one lesson.

The next two lessons are about problems that ask students to find the biggest possible/smallest possible amount of something. This type of problems pop up all over the place, and it is sad that many people make silly mistakes and do not know that a correct solution for such problems consists of two parts. We will talk about it and give some quite hard problems to practice.

Topics

Course 3-A

Prerequisites

The number one prerequisite is having good enough algebraic skills. Variables should no longer scare the student and things like *ax + bx *= (*a + b*)*x* should also look easy. Also, it is recommended to already know what mathematical games are (we talked about them in the beginning of 1-A). Finally, just like for 2-B, we need students to have problem-solving experience so that we do not spend too much time on explaining what a proof is. In particular, students should know what a proof by contradiction is.

About the content

The first week is a well-known fact that is easy to prove by contradiction. But even though it is easy, there are thousand of problems where this fact helps.

The next three lessons are dedicated to the beginnings of algebra and number theory. The key point is to introduce divisibility and remainders and see how they help solving a wide-range of different problems. We will also quickly talk about formal proofs of seemingly-obvious facts since in mathematiccs, taking things for granted is taboo. Ability to prove things from first principles is vital for good understanding of what is going on and how to apply it!

The last two lessons are about mathematical games and in particular about certain type of winning strategies. In the last lesson we will talk about mathematical games of certain type in general and see a cool trick to prove that some player has a winning strategy without actually finding the strategy :)

The next three lessons are dedicated to the beginnings of algebra and number theory. The key point is to introduce divisibility and remainders and see how they help solving a wide-range of different problems. We will also quickly talk about formal proofs of seemingly-obvious facts since in mathematiccs, taking things for granted is taboo. Ability to prove things from first principles is vital for good understanding of what is going on and how to apply it!

The last two lessons are about mathematical games and in particular about certain type of winning strategies. In the last lesson we will talk about mathematical games of certain type in general and see a cool trick to prove that some player has a winning strategy without actually finding the strategy :)

Topics

Course 3-B

Prerequisites

3-A and the same level of algebraic skills are the key prerequisites for this one. Moreover, it would be nice if the student was not scared of things like "7 cube" (i.e dealing with exponents). Also, please revise whatever it was taught about areas in schools, we will need some of this knowledge for the last lesson.

About the content

The first two lessons are mostly algebraic. Actually, problems on decimal representation are perhaps the most common number-theoretical problems in maths olympiads for students of age 11-13 (because other number-theory requires much more knowledge). So it is worth talking about it.

Then we will talk about graph theory, a subject that is unfairly neglected by educational institutions in many countries. The set up is easy and moreover graph theory is one of the branches of mathematics that has tons of real-life applications, especially in IT. In this one lesson we will start from the very beginning, but as always there will be a number of interesting problems in the extra sheet.

Next: invariants and monovariants. Both are simple ideas that are useful for olympiad-style problems and in serious mathematics later. It is likely that most of the people already know the idea, but it is not likely that many students can solve all of the problems we will give on this one idea :)

Finally, we will talk about a cool simple formula for evaluating the area of figures on a grid, known as Pick's formula. Moreover, we will prove this formula and give some problems on it.

Then we will talk about graph theory, a subject that is unfairly neglected by educational institutions in many countries. The set up is easy and moreover graph theory is one of the branches of mathematics that has tons of real-life applications, especially in IT. In this one lesson we will start from the very beginning, but as always there will be a number of interesting problems in the extra sheet.

Next: invariants and monovariants. Both are simple ideas that are useful for olympiad-style problems and in serious mathematics later. It is likely that most of the people already know the idea, but it is not likely that many students can solve all of the problems we will give on this one idea :)

Finally, we will talk about a cool simple formula for evaluating the area of figures on a grid, known as Pick's formula. Moreover, we will prove this formula and give some problems on it.

Topics