In this post I am going to write about my favourite books from a series of excellent problem solving books called the Math Circles Library. I will give you a flavour of each book, and an example problem from each that you might enjoy trying to solve. If you are a teacher, student, or just like solving problems, you should definitely invest in these books!

As a teacher, perhaps my favourite book in the series is

In the book, the author aims to find out how and what to teach small children, but the questions he asks (and tries to answer) are hugely relevant for teachers of children of all ages. He is extremely honest when reflecting on what worked and what didn't, and there are moments of true joy and despair (here is a typical section). It is a beautifully written book, and contains a great deal of insight into the teaching of maths.

The book also contains a number of extremely creative and interesting ideas for teaching that would be suitable for KS1-3, and could be adapted for older students. Here is a typical 'game':

One of the boys (the 'executioner') leaves the room, while the other one (the 'condemned') gets 4 white and 4 black balls. He then distributes them into two boxes any way he wants. The executioner comes back, randomly selects a box, and draws a ball from it (also at random). A black ball means death and a white means freedom.

What is the optimal strategy? What is the probability of freedom?

**Math from Three to Seven**(Zvonkin). This book is the author's diary of a maths circle he held with his son and three of his friends aged 3-4. .In the book, the author aims to find out how and what to teach small children, but the questions he asks (and tries to answer) are hugely relevant for teachers of children of all ages. He is extremely honest when reflecting on what worked and what didn't, and there are moments of true joy and despair (here is a typical section). It is a beautifully written book, and contains a great deal of insight into the teaching of maths.

The book also contains a number of extremely creative and interesting ideas for teaching that would be suitable for KS1-3, and could be adapted for older students. Here is a typical 'game':

One of the boys (the 'executioner') leaves the room, while the other one (the 'condemned') gets 4 white and 4 black balls. He then distributes them into two boxes any way he wants. The executioner comes back, randomly selects a box, and draws a ball from it (also at random). A black ball means death and a white means freedom.

What is the optimal strategy? What is the probability of freedom?

The next pair of books I want to talk about is

15 chairs are placed around a circular table. On the table are name cards for 15 guests. After the guests sit down, it turns out that none of them are sitting in front of their own card. Prove that the table can be rotated so that at least 2 guests are correctly seated.

Can you see how to solve this problem? [Hint: use the pigeonhole principle]

Apart from the rich mathematical ideas at play, this book has inspired me to make a new year's resolution: to create a maths circle for the schools in London...

**A Decade of the Berkeley Math Circle:****Volumes 1**and**2**. The mathematical content in each book is more advanced than the book above (suitable for able students at KS4-5). Each section contains lots of contest-style problems that use a particular mathematical technique, together with full notes on how to go about solving them. Here is a typical problem from Volume 1:15 chairs are placed around a circular table. On the table are name cards for 15 guests. After the guests sit down, it turns out that none of them are sitting in front of their own card. Prove that the table can be rotated so that at least 2 guests are correctly seated.

Can you see how to solve this problem? [Hint: use the pigeonhole principle]

Apart from the rich mathematical ideas at play, this book has inspired me to make a new year's resolution: to create a maths circle for the schools in London...

**Any London teachers reading this: if you're interested in setting up a maths circle, let me know!**The next book,

Basil has a plastic triangle with 30, 60 and 90 degree angles and no marks on it. He can trace the triangle, or he can place a part of or an entire side of the triangle along a previously drawn line and trace the other sides of the triangle. He needs to draw a 15 degree angle. How can this be done with only the triangle?

**Invitation to a Mathematical Festival**(Yashchenko), contains problem sets given to school students at the Moscow Mathematical Festival over a number of years. It has lots of interesting problems to solve with full hints and solutions. Many of the problems do not require a high level of pre-requisite knowledge, focusing instead on problem solving skills - most of the problems are suitable for students from KS3 upwards. Here is an example of a typical problem:Basil has a plastic triangle with 30, 60 and 90 degree angles and no marks on it. He can trace the triangle, or he can place a part of or an entire side of the triangle along a previously drawn line and trace the other sides of the triangle. He needs to draw a 15 degree angle. How can this be done with only the triangle?

In a similar vein is the book

Can you solve it? Is the same true with three colours? Does it have to be a circle? Why an isosceles triangle - will there be an equilateral triangle?

**Moscow Math Circle**(Dorichenko) which gave me the inspiration to write my own problem sets on this website. There are lots more problem sets in the book (much better than mine), with a range of questions again suitable for KS3-5 students - I particularly like the additional problems, which could form starting points for some nice investigations. Here is an example:*A circle has been coloured with two colours. Prove that there will be three points of the same colour that form an isosceles triangle.*Can you solve it? Is the same true with three colours? Does it have to be a circle? Why an isosceles triangle - will there be an equilateral triangle?

To challenge the most able students (and teachers), the books

Here is an example question:

Four grasshoppers sit at the vertices of a square. Every now and then, one of them hops over another, landing at a point symmetric, with respect to the jumped-over grasshopper, to where it jumped from [i.e. it jumps the same distance and direction to and from the jumped-over grasshopper]. Prove that at no time can the grasshoppers occupy the vertices of a bigger square than the original one.

If this hasn't driven you mad enough, try this similar problem that springs to mind:

Three frogs are placed on three vertices of a square. Every minute, one frog leaps over another frog, in such a way that the "leapee" is at the midpoint of the line segment whose endpoints are the starting and ending position of the "leaper." Will a frog ever occupy the vertex of the square that was originally un-occupied?

**Moscow Mathematical Olympiads 1993-1999**and**2000-2005**contain a number of (some very difficult) problems, again with full hints and solutions. Some of the problems could conceivably be accessed by KS3-4 students, but is probably more suitable for students in KS5.Here is an example question:

Four grasshoppers sit at the vertices of a square. Every now and then, one of them hops over another, landing at a point symmetric, with respect to the jumped-over grasshopper, to where it jumped from [i.e. it jumps the same distance and direction to and from the jumped-over grasshopper]. Prove that at no time can the grasshoppers occupy the vertices of a bigger square than the original one.

If this hasn't driven you mad enough, try this similar problem that springs to mind:

Three frogs are placed on three vertices of a square. Every minute, one frog leaps over another frog, in such a way that the "leapee" is at the midpoint of the line segment whose endpoints are the starting and ending position of the "leaper." Will a frog ever occupy the vertex of the square that was originally un-occupied?

I hope you enjoy these books; in future posts I will be writing about other maths books in my collection.