Sunday, 29 May 2011

Greenwich Maths Challenge 6

Here is the sixth Greenwich Maths Challenge. As usual there will be a small prize for the first correct solution emailed to by a Greenwich undergraduate.

If you specify the value of y at n different values of x there is a unique polynomial y=p(x) of degree n-1 which takes these values. But suppose I have a polynomial p(x) of unspecified degree, whose coefficients are all non-negative integers. If you give me a value of x I will tell you the value of p(x). What is the minimum number of evaluations you need to make, to be able to identify all the coefficients of my polynomial? To win the prize you must justify your answer.

Friday, 20 May 2011

Greenwich Maths Talent 2011

Greenwich MathSoc logoa graduation showcase supported by the Institute of Mathematics and its Applications (IMA).

King William Court 315, University of Greenwich, Friday 27 May, 6 - 7:30pm

Cheese, wine and soft drinks will be served.

This is an opportuty to celebrate the success of graduating maths students and to find out about the exciting projects they have worked on during the final year.

The showcase will be opened by Professor Tom Barnes, Deputy Vice-Chancellor.

The twitter hashtag is #GMT2011. There is also an event website.

All welcome - if you are coming, please tell Noel-Ann Bradshaw ( for catering purposes.

Wednesday, 27 April 2011

Two new books (and three old ones)

Biology is a big application area of mathematics at the moment, and here are two new books which I am looking forward to reading this summer:

Ian Stewart, Mathematics of LifeMartin Nowak, Super Cooperators
Stewart says that

"Mathematical theory and practice have always gone hand in hand, from the time primitive humans scratched marks on bones to record the phases of the Moon to the current search for the Higgs boson using the Large Hadron Collider. Isaac Newton's calculus informed us about the heavens, and over the past three centuries its successors have opened up the whole of mathematical physics: heat, light, sound, fluid mechanics, and later relativity and quantum theory. Mathematical thinking has become the central paradigm of the physical sciences.

"Until very recently, the life sciences were different. There, mathematics was at best a servant. It was used to perform routine calculations and to test the significance of statistical patterns in data. It didn't contribute much conceptual insight or understanding. Most of the time, it might as well not have existed.

"Today, this picture is changing. Modern discoveries in biology have opened up a host of important questions, and many of them are unlikely to be answered without significant mathematical input, The variety of matheamtical ideas now being used in the life sciences is enormous, and the demands of biology are stimulating the creation of entirely new mathematics, specifically aimed at living processes. Today's mathematicians and biologists are working together on some of the most difficult scientific problems that the human race has ever tackled - including the nature and origin of life itself.

"Biology will be the great mathematical frontier of the twenty-first century."

Curiously enough, three of the most exciting maths books I have come across, and which have influenced and inspired my teaching, as students will have noticed, also relate to the mathematics of life.

Robert Axelrod, The Evolution of Co-operation
D'ARcy Thopmpson, On Growth and Form
Matt Ridley, The Origins of Virtue

Big Noise

Simon singh's Enigma machine simulator

No, not one of our first year lectures, but the Bexley Big Noise STEM Careers Fair. This was an event for year 9 and 10 students in Bexley to promote careers in science, engineering, technology and maths. Kevin and I went with two of our students (thanks Ameli and James) , and we had a wonderful time, demonstrating code-breaking, mathematical modelling and actuarial mathematics, to lively and enthusiastic mathematicians of the future.

Saturday, 9 April 2011

Young Operational Research Society Conference

Last Monday I travelled to Nottingham to the Young Operational Research Society Conference.

My main reason for attending was to present a paper on my research concerning a new Multi-Objective Evolutionary Algorithm for Portfolio Optimisation. This took place on the Tuesday Morning and was well received.

However of even greater benefit was the information I picked up from those working in OR about the skills they needed graduates to have. Nowadays it is important that, as well as having good technical skills, graduates also need good interpersonal skills and business awareness. Much of this can be acquired through the group projects we set at Greenwich and our focus, in the second year, on employability.

Many of the people I spoke to were involved in building simulation models. Something I will emphasise when I teach this next year. I took part in a demonstration of a software called SIMUL8 (we will be using this in OR next year). We had to simulate a nightclub and despite the fact that I have never been to one and do not know what Indie music is our team won!

More information about the OR Society can be obtained from:

Friday, 18 March 2011

MathSoc's Pi Day Party

On 14th March (3.14) the maths staff and students celebrated pi day. We had a short talk on the history of pi which included learning some fun mnemonics like 'How I Need A Drink Alcoholic Of Course' (3.1415926) and having a competition to see who could memorise the most digits of pi. This was won by Nic Mortimer (3rd year - BSc Decision Science) who memorised 101. Karen Richardson from the library came a close second.

We also listened to a pi song - available here:

This was followed by some pi pie and pi cake made by Noel-Ann Bradshaw, Ameli Gottstein (3rd year - BSc Maths) and James Howe (3rd year - BSc Maths). Pictures by Michael Dullaway (3rd year - BSc Maths).

Greenwich Christmas Quiz

Congratulations to Ameli Gottstein who won this year's Greenwich Christmas Maths Challenge. This completes a clean sweep for Ameli who has won in each of her three years as a Greenwich undergraduate!

Here are solutions to the questions.

Q1. Identify the mathematicians whose names are given below as anagrams. Accents and punctuation marks such as hyphens are omitted, and spellings are taken from the MacTutor History of Mathematics website


Answers: (a) Wren, (b) Kelvin, (c) Kepler, (d) Jia Xian, (e) Hamilton, (f) Bernoulli, (g) Archimedes, (h) Von Neumann, (i) Kolmogorov, (j) Kovalevskaya

For information on these mathematicians see the MacTutor website.

2. A ship is at anchor in a harbour. A spectator sees a ladder dangling from her deck. The bottom four rungs of the ladder are submerged, each rung is two centimetres wide and the rungs are eleven centimetres apart. The tide is rising by eighteen centimetres per hour. After two hours, how many rungs will be submerged?

Answer: still only four rungs. The ship and ladder rise with the tide!

Q3. For Spanish, Russian or Hebrew, it’s 1. For German, it’s 7. For French, it’s 14. What is it for English?

Answer: The answer is 7. It’s the first integer whose name in the language has more than one syllable.

Q4. How many people is "Twice two pairs of twins"?

Answer: Eight

Q5. Each integer from 1 to 10^10 (ten billion) is written out in full (for example 211 would be "two hundred eleven" and 1042 would be "one thousand forty two" - the word "and" is not used), and the numbers are then listed in alphabetical order (ignoring spaces and hyphens). What is the first odd number to appear in the list?

The book answer is 8,018, 018,885 (eight billion, eighteen million, eighteen thousand, eight hundred eighty five).

Q6. The Ruritanian National Library contains more books than any single book on its shelves contains words. No two of its books contain the same number of words. Can you say how many words are contained in one of its books?

Answer: At least one book contains zero words. Use the Pigeon-hole Principle!

Q7. A maths lecturer has a collection of eighteenth-century mathematical pornography in two bookcases in a room 9 by 12 feet (a foot is an archaic unit of measure about 30cm long). Bookcase AB is 8.5 feet long and bookcase CD is 4.5 feet long. The bookcases are positioned centrally on each wall and are one inch from the wall, as shown in the diagram.


Some students are going to visit the lecturer and she wishes to protect the students from the books and vice versa. The lecturer decides that the best way to do this is to turn around the two bookcases so that each is in its starting position but with the ends reversed so that the books are all facing the wall. The bookcases are so heavy that the only way to move them is to keep one end on the floor as a pivot while the other end is swung in a circular arc. The bookcases are so narrow that for the purpose of this problem we can consider them as straight line segments. What is the minimum number of swings required to reverse the two bookcases?

(For more about eighteenth-century mathematical pornography, see Patricia Fara's recent Gresham College lecture.)

Answer: Eight swings are enough. For example, (1) Swing end B clockwise 90 degrees; (2) swing A clockwise 30 degrees; (3) swing B anti-clockwise 60 degrees; (4) swing A clockwise 30 degrees; (5) swing B clockwise 90 degrees; (6) swing C clockwise 60 degrees; (7) Swing D anti-ckockwise 300 degrees; (8) swing C clockwise 60 degrees.

Q8. The number 2 to the power 29 has nine digits, all different: which digit is missing? (Calculator not required.)

The answer is 4.

The way to do this without a calculator is to use the fact that, if you divide a number by 9, the remainder you get is the same as the remainder when you divide the sum of its digits by 9. (This is the basis of lots of maths tricks you can impress your friends with.) For example, 123456 divided by 9 has remainder 3; so does 1+2+3+4+5+6 = 21.

Consider the powers of 2 divided by 9 (draw your own table). It’s easy to see that the pattern repeats every six powers. (You may know Euler’s Theorem relatign to this.) So 229 will have the same remainder divided by 9 as 25 does: this is, 5, which must be the sum of the nine digits. Now the sum of all the ten possible digits 1+2+3+…+9+0 is 45, so the sum of eight of them is between 36 and 45, and the only number in that range which leaves remainder 5 when divided by 9 is 41, which means 4 must be left out.

Q9. What is the 99th digit to the right of the decimal point in the decimal expansion of (1+√2) to the power 500? (Calculator not required.) (In case this isn't displayed correctly by your browser, the expression is (1+sqrt(2))^500, where sqrt(2) means the positive square root of 2.

The answer is 9.

At first sight this seems an impossible difficult question without doing detailed calculations to a huge number of decimal places. But consider the following.
Let x = (1+√2) and x’ = (1-√2). What happens when we add powers of x and x’?
Let y = (1+√2)^500 + (1-√2)^500

We can use the binomial theorem to expand the two powers. Now, for odd powers of √2, the coefficients will have opposite signs in the expansions of x^500 and x’^500 and they will cancel, so we have an expression for y in terms of even powers of √2. But an even power of √2 is a power of 2, so it is an integer. So y is an integer.

Now, what is x’^500? Well, x’ = 1-√2 is about -.414: it’s a negative number with absolute value less than ½. So the 500th power of x’ is positive and is less than (1/2)^500, which is very small – about 4x10^-192. So the decimal expansion of x’ begins with more than 100 zeroes.

So x plus a positive number beginning with 100 zeroes gives an integer: so the first 99 digits after the decimal point in x must all be 9s.

When I first saw this problem I thought it was impossible so I looked at the answer straightaway. But having read the answer it doesn’t seem so impossible – because when one has a term like (1+√2) in an expression it’s a common trick to think about (1-√2). It’s hard to see any other way one could tackle this problem! So having looked at the answer, I’m now sure I could have done it myself if I’d bothered. (I may be being over-optimistic!)

Q10. Here are two messages enciphered using substitution ciphers. What do they say?



(b) was designed to avoid the letter E to make frequency analysis a little harder.

Questions 3, 4, 8 and 9 came from Mathematical Mind-Benders by Peter Winkler and Question 5 from the same author’s Mathematical Puzzles: A Connoisseur’s Collection. Questions 2, 6 and 7 are from David Wells's The Penguin Book of Curious and Interesting Puzzles.