At the Greenwich University Maths Day on 5 July 2018 Rob Eastaway showed students the following game. Everyone stands up and chooses either heads or tails. Rob tosses a coin. Those who guess wrong sit down. Those left standing choose again, toss again, wrong-guessers sit down again – through round after round until only one person (or none) is left.

What is E_{n}, the expected value of the number of tosses, when you start with n people, to leave everybody, or everybody bar one, sitting? At first sight you might think this is log_{2}n, because each toss roughly halves the number standing. But that is wrong.

Click here for a calculation. Thanks to Zhaoqi Chen for help with this.

]]>Robert Langlands has been awarded the 2018 Abel Prize, one of the chief honours in the world of maths, for opening up a research program which has dominated large areas of maths for the last 50 years, and still has many unsolved problems.

Langlands is Canadian by origin, but has lived and worked in the USA for many years. Another honour he has received is being allocated Einstein’s old office in the Institute for Advanced Study near Princeton University.

Ed Frenkel’s book “Love and Math”, alongside being an autobiography, tries to explain the Langlands program. Despite its breadth, the Langlands program is essentially about connections between different areas of maths at a level not studied at school, or even in most first-degree maths course, so a quick explanation is difficult. (Click here for more.)

This report by Rachel Thomas from Plus magazine does a better job than any other short explanation I’ve seen.

In 1967 a 30-year-old mathematician, still in the early stages of his career, wrote a 17 page letter to the eminent French mathematician André Weil. The covering note that he sent with the letter said:

“After I wrote [this letter] I realized there was hardly a statement in it of which I was certain. If you are willing to read it as pure speculation I would appreciate that; if not – I am sure you have a waste basket handy.”

The letter may have contained many unproved statements, but these turned out to be asking questions that would create a whole area of mathematical research, unmatched in modern mathematics for its scope, deep results and sheer size in terms of the number of mathematicians it has enticed. The author of the letter – Robert Langlands – has been awarded the 2018 Abel Prize for his “visionary program” that bridges previously unconnected areas of mathematics, and is now frequently described as a “grand unified theory of mathematics”.

The insights contained in Langlands’ letter to Weil, and more fully in his 1970 lecture Problems in the Theory of Automorphic Forms, set out a programme of research that is now referred to as the Langlands program. Langlands made a number of conjectures about the connections between two previously unrelated areas: number theory and harmonic analysis.

Harmonic analysis explores how functions and signals can be represented as a sum of waves. For example, the sound wave of the middle A on a tuning fork is a perfect example of a sine wave, written mathematically as sin(x)… Automorphic forms are a more generalised version of periodic waves, like the familiar sine wave, which operate in more complicated geometric settings….

]]>Answer sheet (some of you did better on the shortcuts than the official answer sheet).

]]>*Above: the memorial to the great 20th century mathematician David Hilbert, in Göttingen, Germany, carries Hilbert’s words: “Wir müssen wissen. Wir werden wissen”. “We must know. We will know”.*

*Above are some of those already selected. I’ll regularly update the full list of titles selected, below. Some students or ex-students have just asked me to pick one for them.*

*Adam Spencer’s Book of Numbers*: Rebecca Lydon

*Are you smart enough to work at Google?*: David Trieu

*As easy as pi*: Sunneth Lawrence

*The Beauty of Geometry*: Alex On

*The Black Swan*: Michelle Villamagua

*A Book of Abstract Algebra*: Deniz Yukselir

*A Cartoon Guide to Statistics*: Lou-Lou Batchelor

*The Code Book*: Jeffrey Sylvester

*The Colossal Book of Mathematics*: Megan Francis

*A Course of Pure Mathematics*: Chloe Harper

*As easy as pi*: Sunneth Lawrence

*e – The Story of a Number*: William Bassoumba

*Elementary Number Theory*: Seun Tijani

*Fascinating Fibonaccis*: Abdal Mohammad

*Fermat’s Last Theorem*: Kelly Ung

*Fifty Mathematical Ideas*: Genie Louis de Canonville

*Finding Moonshine*: William Ginzo

*A Friendly Introduction to Number Theory*: Connie Tooze

*Gödel, Escher, Bach*: Tobi Adebari

*How Many Socks Make a Pair?*: Susan Okereke

*Hyperspace*: Helen Hoang

*Introduction to Statistical Analysis for Economists*: Rashi Bhatt

*The Irrationals*: Mya Titus

*King of Infinite Space*: Lola Behanzin

*Love and Math* (one copy): Tegan Hill

*Love and Math* (the other copy): Aniqa Hussain

*Mathematics: A Very Short Introduction*: Rose Hemans

*The New Turing Omnibus*: Reece Jackson

*Principia Mathematica*: Mugisha Uwiragiye

*The Problem-Solver’s Handbook*: Conner Lake

*Professor Povey’s Perplexing Problems*: Javonne Porter

*The Secrets of Triangles*: Abass Doumbia

*The Simpsons and Their Amazing Mathematical Secrets*: Bolaji Atanda

*Six Not-So-Easy Pieces*: Danny Ryan

*Street-Fighting Mathematics*: Tim Ekeh

*A Tour of the Calculus*: Ashley Anigboro

*Why Things Are The Way They Are*: Howard Tran

*Some thin volumes whose titles would be hard to read from a pic of the bookshelves. Nelsen’s Proofs Without Words is just what the title says: each page is a diagram which proves a result, maybe in trigonometry, arithmetic, or algebra, just by looking. Terry Tao’s Solving Mathematical Problems is in my view the best book on problem-solving.*

*More thin volumes, including the famous Stanford Mathematics Problems Book.*

*Second from the left, with the title hard to read, is Hardy’s classic Course of Pure Mathematics.*

*To the right of Ronan’s Symmetry and the Monster (i.e. on the left of the run of books) are Marcus du Sautoy’s The Music of the Primes, Durell’s Advanced Algebra (an old textbook, but beautifully clear and concise), and Simon Singh’s Fermat’s Last Theorem.*

*On the left, with the title hard to read, is Freund’s Mathematical Statistics. In the middle, again with the title hard to read, is Serafina Cuomo’s Ancient Mathematics. Newman’s four volumes on The World of Mathematics were my favourite reading when I was a young school student.*

*There is a second copy of Love and Math here. The shelves also include two copies each of Euclid’s Elements (different editions); of Simon Singh’s The Code Book; of David Berlinski’s A Tour of the Calculus; and of Marcus du Sautoy’s Finding Moonshine. The two blue-ish books with titles hard to read are Ian Stewart’s Does God Play Dice? and Fearful Symmetry.*

*Courant and Robbins’ What Is Mathematics (far left) is maybe the most famous general-introduction-to-maths book of all time, and with good reason. My general rule has been to keep for myself the postgraduate-level books, and leave with CoLA the undergraduate-level, school-level, and pop-maths books: Russell and Whitehead’s Principia Mathematica is much more difficult than most postgraduate books, but it is such an important text in the history of maths that I thought I should leave it with CoLA. Some students may remember me showing them page 300-odd, where Russell and Whitehead finally comment: “From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2” https://math.stackexchange.com/questions/278974/prove-that-11-2. The book in the middle with the title difficult to read is Burn’s Numbers and Functions, famous as a guide to “the transition from studying calculus in schools to studying mathematical analysis at university”. David Berlinski’s A Tour of the Calculus may be the most “literary” book about mathematics ever written: it’s a very unusual book, and fascinatingly readable.*

*Chandrasekar’s book is a closely-reasoned account, with very little technical maths, of how quantum effects shape things visible in everyday life.*

*Feynman’s two books give readable introductions to important sections of mathematical physics.*

*Emmy Noether’s main work was in abstract algebra, but her First Theorem was part of mathematical physics. It says that if a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time. Time translation symmetry gives conservation of energy; space translation symmetry gives conservation of momentum; rotation symmetry gives conservation of angular momentum, and so on.*

*The Hilbert Challenge is an account of the 23 problems which David Hilbert set for 20th century maths to solve at the International Congress of Mathematicians in 1900, and what’s happened on them. It has a lot of maths, but is well-written enough that you can follow it even if some of the maths goes over your head.*

*Silverman’s book on number theory written for an American university course designed to attract students whose “majors” are in other fields but have a little school maths background, and an interest in expanding their cultural knowledge to include maths.*

**Maths prize puzzle for 21 July 2018 cake numbers**

In three dimensions, what is the maximum number of pieces you can cut a cake into with one cut? Two cuts? Three cuts?

What is the general formula for n cuts?

You may find it useful to think of the “pizza numbers” – maximum number of pieces you can get from a pizza, in two dimensions, with n cuts: 1, 2, 4, 7, 11, 16, 22, 29… (bit.ly/pizza-n).

]]>F(n) and M(n) are defined in a recursive, or doubling-back-on-themselves, way as:

F(n) = n─M(F(n─1))

M(n) = n─F(M(n─1))

F(0) = 1; M(0) = 0.

• By using a spreadsheet program with VLOOKUP, or by hand, calculate values of F and M up to n=55.

For most n, F(n)=M(n).

• What can you find about when F(n)≠M(n)?

Here are the first rows of the spreadsheet:

Longer list of F at http://oeis.org/A005378

and of M at http://oeis.org/A005379

**F(n) is not equal to M(n) if and only if n+1 is a Fibonacci number.**

The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… – each number formed by adding the two previous ones in the sequence.

]]>Calculating a Minimum Spanning Tree and multiplying its total weight (length) by 2 is a quick way to get an upper bound for the Travelling Salesman Problem.

You can do a tour of all the vertices by going “out” along the Minimum Spanning Tree and then returning along each and every branch to whatever vertex you choose to start at.

Usually, if you look at the graph you can see a couple of obvious shortcuts for the “return journeys”. In this example, you can avoid having to go along DE and EG twice by returning from G to D by the edge of length 1102. You can avoid having to go along AC, CD, and BD twice by returning to B by the edge of length 982.

Edexcel will want you only to find one or two shortcuts, and to leave all the “outward” journeys along the Minimum Spanning Tree as they are.

So, oddly, you need to guard against being “too good”. You actually know for this graph that you can do better by cutting out journeys along DE and DB, even one way, altogether, and going

A → C → D → G → D → F → B → A

But don’t do that in an exam! Since the markers aren’t as smart as you are, they will probably give you zero marks. Leave the “outward” journeys along the Minimum Spanning Tree as they are, and limit yourself to finding one or two shortcuts for the “return” journeys.

Here are two examples from Ex.5B in the book.

Working with the full graph can be complicated, especially if you’re given it as a table rather than a diagram, so you may do better to sketch just the Minimum Spanning Tree then cross-check with the table or the full diagram for shortcuts.

]]>The travelling salesman problem (TSP) is: find the shortest *tour*, returning to your start, which visits each vertex at least once.

**CLASSICAL AND PRACTICAL**

Anyway, that’s what the book calls the “practical travelling salesman problem”. The book calls the problem of finding a shortest tour visiting each vertex *exactly* once the “classical travelling salesman problem”.

All you need to know for now about the “classical travelling salesman problem” is how to answer the question: what is the difference between the “classical travelling salesman problem” and the “practical travelling salesman problem”. All our algorithms deal with the practical TSP.

**Complete Table of Least Distances**

Because we’re dealing with the practical TSP, we’re interested only in the shortest route between any two vertices, and we aren’t bothered if that route means going through a vertex we may already have visited.

Instead of working with the original graph, we work with a *complete table of least distances*, which shows the shortest distances between vertices (even if they’re indirect, and even if that means ignoring direct routes which are longer).

**Upper bounds and lower bounds**

As with Bin-Packing, no algorithm guarantees you an *exact* answer for the TSP (except a checking-every-route one which is impossibly long-winded for even a moderate number of vertices, e.g. for just 20 vertices it means checking 1.2×10^{18} possibilities, several years’ computing time on a fast modern computer).

The algorithms give you upper bounds and lower bounds. The best upper bound is the smallest one. The best lower bound is the biggest one.

**Nearest neighbour**

Nearest neighbour says: 1. Pick a vertex to start. (For A level, the question will tell you which one to pick). 2. Work from there, step by step, at each step adding the “nearest neighbour” vertex to your last vertex. 3. Finally, connect back to start.

It’s like Prim, except that at each stage you choose the “nearest neighbour” route from *your last vertex*, rather than from *any vertex you’ve reached so far*.

Problem with nearest neighbour is that it may give you a really long distance to connect back from the last vertex to the start. Still, it gives you an upper bound.

**MST×2**

MST×2 gives you, not a better upper bound necessarily, but a quicker-to-calculate upper bound.

Find a minimum spanning tree (MST), using Kruskal or Prim. Then going out along that MST, and getting back to the start by retracing your steps from each “branch”, is a tour and reaches all vertices.

So MST×2 gives an upper bound.

**MST×2 with shortcuts**

MST×2 gives an upper bound, but maybe a rubbish one. Often you can improve on it just by a bit of human intelligence. For example, choose a start point on your MST and look at the most distant branches. There may be “shortcuts”, routes back to the start from the ends of those most distant branches, which are quicker than retracing your step through the whole MST.

**MST and RMST**

Just MST gives you a lower bound, because the MST tells you the shortest total distance to visit all vertices. Pretty much always the MST won’t be a tour, but any tour visiting all vertices must be at least as long.

That is a lower bound. But it is so rubbish that the textbook doesn’t even mention it.

Instead it mentions RMST, Residual Minimum Spanning Tree, which is the MST method improved by a bit of human intelligence.

Take out any vertex. (For A level, the question will tell you which one to take out). Find an MST for all the *remaining* vertices. Then find the two shortest routes which connect that taken-out vertex back in with the MST for those remaining vertices.

That is a lower bound.

If the route you get by doing RMST is a Hamiltonian cycle (meaning: it’s a tour which goes through each vertex exactly once), then that route is not just a lower bound. It’s an exact answer. But usually the route you get by doing RMST is not a Hamiltonian cycle. It gives you not an exact answer but a lower bound.

The video is good but concentrates much more on the “classical” TSP. Also, it’s not quite right about trial-and-improvement. Unintelligent trial-and-improvement, trying absolutely every possibility, becomes unworkable even with six vertices, but intelligent trial-and-improvement is really quite good for fairly small numbers of vertices. ]]>