Try our Bill Tutte-inspired Maths Puzzle

Try solving this puzzle from Trinity mathematicians, inspired by Trinity alumnus and Second World War codebreaker Bill Tutte.

Bill Tutte, who arrived at Trinity in 1935 to study Natural Sciences, specializing in Chemistry, was keenly interested in maths.

Later in life, long after his remarkable achievement cracking the Germans’ Lorenz cipher, he attributed his selection for Bletchley Park to his ability to solve mathematical puzzles.

Together with three friends, all studying maths, Tutte solved a mathematical conundrum – the squaring of the square – which was thought to be impossible. Listen to Trinity Fellow Professor Bela Bollobas explaining the significance of this achievement.

Tutte is included in the Royal Mail’s Valour and Victory set of stamps to mark the 80^th anniversary of VE Day.

This puzzle created by Trinity mathematicians is definitely less high-pressured than figuring out highly encrypted top-secret messages.

‘There is no doubt this is testing,’ says Trinity Fellow in Mathematics Professor Imre Leader. ‘But if you are good at thinking mathematically and logically, give it a try.’

Open to those 16 and under in the UK. Email your answer to: communications@trin.cam.ac.uk by 5pm Friday 16 May.

The prize is a free guided tour of Trinity College Cambridge for the winner and one guest. Trinity’s Porters guide visitors around the College throughout the year on pre-booked tours.

The competition is free to enter.

Small print: the prize winner will be announced on Trinity’s social media channels. The Porter-led tour must be booked in advance and be taken in 2025. There is no monetary-equivalent prize.

The puzzle

We have some tasks that need to be performed, each one taking a day. Some pairs cannot be done on the same day (maybe because they need the same piece of equipment, or whatever).

For example, if we had tasks A,B,C,D and we were not allowed to schedule A and B on the same day, and also not allowed to schedule A and C on the same day, then we could easily complete all the tasks in 2 days: do task A on the first day and the rest on the second day.

If we were asked: ‘Is it always true that any set of tasks can be performed in 10 days,’ we would say: ‘Of course not – just pick 11 tasks and insist that no pair can be performed on the same day’.

Easy.

However, in that example we can find three tasks, no pair of which can be performed on the same day. Indeed, any three tasks have this property, in our example.

Calling such a triple of tasks a bad triple, here is the puzzle:

Can you find a family of tasks, and some pairs that cannot be done on the same day, such that this family has no bad triple at all and yet the tasks cannot all be completed within 10 days?

If Trinity’s Mathematics Puzzle has whetted your appetite, in September this year you can participate in the Southampton University National Cipher Challenge. The online competition offers young people the chance to try codebreaking and insights into the cyber world.

Professor Graham Niblo, of the School of Mathematical Sciences at the University of Southampton, said:

This annual online competition recruits ‘agents’ from schools across the UK and introduces them to the thrill of codebreaking, sharpening their skills and knowledge and showing them how the maths and computing they have learned can be used in the cyber world.

The challenge is open to anyone but only full-time students at school in the UK qualify for prizes. The prize giving takes place each year at Bletchley Park, the Second World War codebreaking centre.

Registration for the National Cipher Challenge opens on 8 September https://www.cipherchallenge.org

Professor Niblo said students across the UK take part, ‘many of whom find a new love of maths and computing, as the challenges take them out of their comfort zone and show them just how much they can achieve.’

This year marks the 20^th anniversary of Trinity College’s support of the National Cipher Challenge, which is also sponsored by GCHQ and the Engineering and Physical Sciences Research Council, among others.