Maths Circle

We hosted approximately thirty Year 10 pupils from partner schools in London (including Fulham Boys School, Grey Coat Hospital, Christ’s School, St Paul’s Girls School, and West London Free School).

In addition to the mathematics sessions, break-out activities included a bridge-building engineering challenge in which teams of six pupils constructed a load-bearing bridge exclusively out of A3 paper, lollipop sticks and Sellotape, spanning 50cm, which was then tested – to destruction – by incrementally adding more and more weight for the bridge to bear. The pupils also took on a cypher challenge, again in teams of six, to construct a cypher using fifty 1p coins and fifteen 2p coins, and then – in half-teams – to encrypt and decrypt as much plaintext as possible. Here are some of the activities on offer:

Mathematical Proof (a report by Mr Cullen-Hewitt)

What constitutes a proof? Images of dusty chalkboards heavily laden with obscure symbols and inscrutable diagrams might spring to mind, but in fact we engage in the act of mathematical proof whenever we justify our answers, and some of the best examples can often be summarised in a few short sentences or an elegant diagram. In this session we explored several types of proof, from verbal justifications, to algebraic proofs, to proofs without words. In the process we performed some mathematical magic tricks with multiples of 11, saw three different proofs of Pythagoras’ Theorem, found a formula for the triangular numbers and finished with a delightful derivation of the sum of cubes formula.

The Pythagoras' Theorem proofs: https://nrich.maths.or
g/problems/pythagoras-proofs

What Comes Next? (a report by Mr Morris)

The students started out with the easiest sequence of all: 1,2,3,4,5,... but quickly delved into some trickier problems on linear sequences. In particular a question about finding primes in arithmetic progressions. (The general problem was only resolved in 2004 by Green and Tao). 

Then they looked at quadratic sequences (sequences where the second-differences are constant), as well as some tricky problems related to these. Some groups had already seen how they related to sums of sequences of the first types, whilst other groups rediscovered this fact. A few groups looked at sequences of the form n^2+n+p, and whilst they didn't get a chance to see the full beauty of the algebraic number theory underpinning these sequences, they did spot some interesting patterns and made some nice deductions. 

They also had a chance to look at Fibonacci-type sequences and how they related to the sum of two geometric sequences - a few groups looked at how these sequences connect to combinatorics. 

Finally, the group wrapped up by looking at the look-and-say sequence, 1, 11, 21, 1211, etc, where we established the rule to go between sequences and proved that 4 cannot appear in the sequence.

Teams Maths Challenge (a report by Mr Druce)

 Students split into teams of 4 and took part in two rounds of the UKMT Team Maths Challenge. The first round was the Group Round, in which teams worked through a group of 10 challenging questions. It was a close contest with most teams within a few points of each other. 

The Shuttle round followed; one pair from each team was required to correctly solve a question before the next pair on the team could start on their question. Answers were passed back and forth, with bonus points given to groups that had successfully completed their questions within the time limit.  It was a good test of precision and teamwork. We completed four iterations of this task in a tense contest. As scores were collated at the end of round, students were aware of what was at stake. I was personally impressed by some students' ability to carefully check their solutions before submitting their final solution, especially under the bonus point time pressure. Ultimately, there was a winner, but a lot of enjoyment was had by all teams, all of which had been competitive. 

For those wanting to try it, free Teams Maths Challenge papers are available from: https://ukmt.org.uk/competition-papers

Bases (a report by Dr Harrison)

Can you double a positive integer? Can you halve a positive integer? Can you add two numbers together? In which case you can do multiplication. Via the “Russian Peasant” algorithm, an introduction to binary representation, powers of 2 and the distributive property, we discovered how modern computer chips perform multiplication using just binary addition.