Challenge Level

Here we collect 10 essential pure mathematics problems to get you thinking before you embark on your degree. They will be very useful to anyone intending to study mathematics at university: they will give you a good mathematical grounding in some of the pure, and possibly unfamiliar, topics likely to arise in your degree course and refine your problem solving skills. You will also find the applied mathematics preparation useful and may be surprised what constitutes applied mathematics at university!

Remember, these problems are designed to make you think and there is not necessarily a 'right' way to do them. Approach them in a thoughtful way; they are hopefully both interesting and stimulating. What questions do they raise in your mind? Where do these questions lead you? Take them to a level that feels comfortable for you.

Finally, once you have done the problems, study the solutions. These will give you additional insights into the problems and the underlying mathematics.

Proof sorter | Get started by reconstructing this classic proof of the irrationality of $\sqrt{2}$, which will be found in all undergraduate courses of mathematics. |

Refine your understanding of logical implication and the all-important $\Rightarrow$ and $\Leftrightarrow$ symbols. | |

Modular fractions | Find out about multiplicative inverses in modular arithmetic. |

Ford Circles | Give yourself an algebraic workout and discover the beauty of the Farey sequence. |

The clue is in the question | Practise constructing a complex proof using the properties of a given system. |

Squareness | Explore some interesting relations and their graphs. |

Complex partial fractions | See how the power of algebra is opened up with the help of complex numbers. |

What is a group | Groups are fundamental: get started with the basics here. |

Cube net | Get started with graph theory and some combinatorics. |

Transformations for 10 | Explore how matrices are used to transform vectors in this critical foundation to the mathematics of linear maps. |

The following articles, interspersed with small problems for you to try will be interesting and useful:

An introduction to complex numbers

Introduction to Number Theory

Modular Arithmetic

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