The Intermediate Maths Challenge is an annual competition run by the UK Maths Trust for 13-16 year olds, and has participation in the hundreds of thousands. Around 500 of the top performers in each of Years 9, 10 and 11 qualify for their respective Olympiad; Cayley for Y9, Hamilton for Y10, MacLaurin for Y11.

Each Olympiad lasts two hours, and contains six problems requiring full written solutions. The problems are designed to provide a stern test to those sitting the paper, and so are typically much harder, and less familiar, than standard curriculum questions. But they serve a purpose long after the competition itself has concluded – they provide teachers, pupils and mathematics enthusiasts alike with interesting, thought-provoking problems to be tackled and discussed in whatever environment and time-frame they choose.

These problems do not appear out of thin air. They are carefully crafted by humans, some of whom find joy in spending hours playing around with mathematical ideas, then tweaking the parameters to produce something polished. These authors are mathematical artists, and their creations are housed in competitions like the UKMT Olympiads, much like pictures are housed in galleries, or artefacts in museums, to be appreciated long after they were dreamed up.

Inventing such a problem is no small feat, and coming up with a ‘good’ problem, whatever that means, is in my experience quite difficult. But I suspect there are many people who are both capable of producing such problems, now or in the future, and who would enjoy the experience immensely.

As Chair of the Cayley, Hamilton and MacLaurin Olympiads Problems Group, I am now looking for enthusiastic volunteers to take up this challenge. There is no application process, and you are free to devote as much or as little time as you wish. If you come up with something you think might be suitable for bright 13-16 year olds, send a word document or a pdf with your problem and a solution to daniel.griller@gmail.com

For the time being, I am reluctant to be more specific in my brief – I don’t want to restrict the creative process in terms of typical categories of problems, or a typical style. I might say more at some point, but presently all I ask is that the problems are original, and that you are an adult; it is possible, however unlikely, that a particularly shrewd teenager might write a problem in the hope that it ends up on an Olympiad that they one day sit themselves!

Feel free to contact me if you would like any more information. And whether or not your inventions end up on the Olympiad papers, I will be very grateful for your efforts; so thank you, and happy composing.

## Problem Solving In GCSE Mathematics

My new book Problem Solving In GCSE Mathematics is now available at Amazon UK, Amazon.com and other Amazon sites.

It is a self-contained workbook, full of challenging problems covering the most demanding parts of the syllabus, with full solutions included.

It should be useful for anyone looking to achieve a top grade at GCSE or IGCSE.

## A little bit of magic

I first came across this problem years ago, somewhere online, but the exact source remains elusive:

Split $\{1, 2, 3, ..., 2n\}$ randomly into two subsets $X$ and $Y$, each containing $n$ integers. Put the elements of $X$ into increasing order $x_1 and put the elements of $Y$ into decreasing order $y_1>y_2>\cdots >y_n$

Prove that $\lvert x_1-y_1\rvert +\lvert x_2-y_2\rvert + \cdots +\lvert x_n-y_n\rvert=n^2$

## Escaping elevens

From a Romanian Selection Test:

What is the greatest possible length of a sequence of consecutive positive integers, so that none of the integers have a digit sum divisible by 11?

## Children and Chocolate

Adapted from a Maths Battle in London:

Several children sit around a circular table, on which lies a collection of 100 chocolates. They proceed to take chocolates, after which it turns out that each child has taken either 6 fewer or 3 times as many chocolates as the child to their left. Prove that some chocolate was left on the table.

## Students

From the Tournament of the Towns:

In a school, more than 90% of the students study both English and French, and more than 90% of the students study English and German. Prove that more than 90% of the students who study both French and German also study English.

## Five People

Here is a problem that appeared in a Maths Battle in London a couple of weeks ago:

Donald, Jack, Peter, Richard and Steven have, in some order, the surnames Donaldson, Jackson, Peterson, Richardson and Stevenson. Donald is 1 year older than Donaldson, Jack is 2 years older than Jackson, Peter is 3 years older than Peterson, and Richard is 4 years older than Richardson. Who out of Steven and Stevenson is older, and by how much?

## The Grid

A PuzzleCritic Original:

An empty 6-by-6 grid of squares is filled with numbers as follows. The first row contains the numbers {1,2,3,4,5,6} in some order. In each subsequent row, the kth number is equal to the position of the number k in the row above. For how many such grids are the first and last rows identical?

## Weights

From the Tournament of the Towns:

A balance and a set of metal weights are given, with no two the same. If any pair of these weights is placed in the left pan of the balance, then it is always possible to counterbalance them with one or several of the remaining weights placed in the right pan. What is the smallest possible number of weights in the set?

## Laundry

A stunner from the Baltic Way:

A family wears clothes of three colours: red, blue and green, with a separate identical laundry bin for each colour. At the beginning of the first week, all the bins are empty. Each week, the family generates 10kg of laundry (the proportion of each colour might vary week to week). The laundry is sorted by colour and placed in the bins. Next, the heaviest bin (only one if there are several) is emptied and its contents washed.

What is the minimum requirement (in kg) of each bin?