Daily Number: 5003
Harewood House, on the northern outskirts of Leeds, is today hosting the Girlguiding UK centenary camp. Harewood was chosen for this event as it was once the home of HRH Princess Mary, who became president of The Girl Guide Association in 1920. Robert Baden-Powel, who founded the Scouting movement in 1907, decided in 1909 that girls should not be allowed to join the same movement as boys. Following that decision, The Girl Guide Association was set up in the UK in 1910.
Not being totally familiar with the Girl Guide eventing calendar, I first learnt of this event from this morning’s TV news. The report predicted choas on the roads around the Harewood House estate because of the imminent arrival of thousands of Guides. What the report didn’t mention was the exact number of Guides it would take to cause chaos on the roads of Yorkshire.
This led me to be curious. I felt a traffic modelling problem forming in my head. Just how many coaches would it take for the Leeds ring road to become unusable?
Sadly, I wasn’t able to find an answer nor did I have time to do sufficient modelling. What I did find, however, was that the best estimate for the number of Guides attending the event is 5,000. I’ve taken this to be a lower bound and so, with a reasonable level of confidence, I think I’m safe to say that there will probably be at least 5,003 UK based Guides present at the Harewood event.
So, with a certain level of predictability, 5,003 is today’s Daily Number. Incidentally, I’m told there may be as many as 2,000 other Guides from around the world attending too but we’ll discount that for now.
So, what’s so interesting about the number 5,003? It is a Sophie Germain prime. That is, 5,003 is a prime number and the number 2p+1, which is 10,007, is also prime.
In number theory, a number p is a Sophie Germain prime if 2p+1 is also prime. The first few Sophie Germain prime numbers are 2 (because 5 is also prime), 3 (because 7 is also prime), 5 (because 11 is also prime) and 11 (because 23 is also prime). The list continues and includes other numbers such as 233 (because 467 is also prime).
These numbers are named after the French mathematician Marie-Sophie Germain. They have interesting applications in Cryptography which I’ll leave to discuss on another day.
Daily Number: 96
At 2315 EDST on 3 August 1958, the USS Nautilus reached the geographic North Pole and became the first vessel to complete a submerged transit across the North Pole.
96 is the number of hours the USS Nautilus spent submerged under the Arctic polar ice cap during its record breaking journey. During that time, it travelled 1,590 nautical miles.
So, our daily number for today is 96. There are a number of interesting things about the number 96. Right now, we’ll focus on the fact that 96 is a Tau number.
A tau number is one which is divisible by the count of its divisors. 96 is divisible by 12 integers: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96. That is, if you divide 96 by one of these numbers, say 6, you get an integer (i.e. a number with no decimal part).
96 is divisible by 12 and, by definition, is therefore a Tau number. Tau numbers are sometimes referred to as Refactorable numbers.
There are 16 Tau numbers up to 100. They are 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96.
Daily Number: 80
80 is the lowest test score made by Pakistan against England.
Needing an unlikely 435 to win the First Test at Trent Bridge, Pakistan were bowled out this morning for 80 runs in 29 overs. As well as being the lowest test score made by Pakistan against England, the score of 80 is now the lowest test match innings for 2010, replacing the 88 made by Australia against the same Pakistan side 10 days ago. What a difference a week makes in cricket.
Aside from my love of cricket, you may be wondering why I’m blogging about this cricket score. Easy – something which happened today is related very clearly to a particular number. As I’m looking for a new challenge I’ve decided to try and do a number challenge.
That challenge is to take a number which is topical and find out something mathematical about that number. So, every couple of days, i’ll take that topical number and briefly explore some properties of that number. Like I say, I’m starting today. Shall we begin?
80 is a Ménage number. The ménage problem considers the number of ways it is possible to seat a set of heterosexual couples around a circular dinner table, such that men and women alternate and no one sits next to their partner. The problem was first formulated in 1891 by Édouard Lucas.
The Ménage sequence begins 0, 1, 2, 13, 80, 579, 4738, 43387, 439792… for 2, 3, 4… sets of couples. 80 is part of this sequence.
Without explaining the maths in detail, because 80 is part of the Ménage sequence and it is the 5th number in that sequence, we make the following conclusion:
If there are 6 couples, there are 80 different ways they can be seated around a circular dinner table so that men and women alternate and nobody is sat next to their partner.
This problem could be solved by working out all the possible arrangements of men and women. The branch of mathematics which deals with problems like this, Combinatorics, has delivered us a number of formulas to help us with problems like these so we don’t have to sit down and work out each and every possible solution. I won’t go into the details here, but, if you are interested, try looking for Touchard’s formula and the 1986 work of Bogart & Doyle.
12th IMA Early Career Mathematicians Conference
I’m just on my way back from Newcastle and the 12th IMA Early Career Mathematicians Conference.
I don’t often get time to sit and think broadly about Mathematics any more. When I do get the chance it’s usually very specific thoughts, to help me broaden my understanding of the fields I’m concentrating on, such as Logic, Computability and Decision Making Processes.
I’ve thoroughly enjoyed today as it gave me a chance to escape from my “usual mathematics” and listen to lectures on different mathematical topics.
I particularly enjoyed Ron Knott’s lecture on Fibonacci numbers, as it reminded me of the early days of my Degree course when we studied them in some detail. The patterns you find in Fibonacci numbers are particularly beautiful and a great signpost to other areas of elegance and beauty within Mathematics.
Lindsay-Marie Armstrong and Sharon Evans gave a talk discussing the differences between a Ph.D route and a route into industry for Mathematicians. I came away from the talk pleased and satisfied with the route I’ve taken so far which has seen me steer past both options to work specifically in the Commercial world. Everyone is different and had my choices and opportunities earlier in life been different then maybe I’d be saying other things right now. That said, I realised that I do enjoy how the application of my Mathematics brings about real change in Commercial fortunes and the results can be seen relatively quickly.
I’ve always been fascinated by the History of Mathematics and how the work of great Mathematicians has changed the face of the world as we know it. I was particularly pleased to see a lecture on the History of Euler on today’s agenda. Robin Wilson’s lecture was fun, very informative and gave a real picture into the life and work of the great man.
The lecture on Dynamic Networks by Nira Chamberlain showed how the topology and nodes of a network can change over time. I came away wondering how some of the concepts discussed in this lecture impact on problems in Game Theory and Probability, such as the Gambler’s Ruin problem, which I consider in my daily Mathematics.
Robin Johnson’s lecture on Single Perturbation Theory reminded me how important it is for a speaker to be passionate about their subject and enjoy what they’re talking about. I haven’t used the methods Robin talked about for nearly 6 years and some of it was a little hazy. Without the enthusiastic and knowledgable delivery of the lecture, I’m not sure I would have dusted off the dark corner of my memory concerned with methods of solving awkward differential equations. It was an impressive demonstration of how to make a very technical lecture interesting and engaging. I’ve taken away a lot from this lecture which I’ll apply to my own teaching and training in the future.
The final lecture was on Magnetohydrodynamics and the Solar Corona by James McLaughlin. This is an area I’ve never looked at before. The meshing together of Electromagnetism with Fluid Dynamics provides an interesting example of how distinct areas of Mathematics cross over to produce new and important results.
It was a great day of Mathematics which has re-energised my appetite for Mathematics away from the problems I study day to day. I’m really pleased I made the effort to travel to the conference and hope it will prove fruitful in the months to come.