Plus Blog

October 2, 2014

We have just learnt a really nice fact about the game of tic-tac-toe. As you may know, tic-tac-toe always results in a draw if both players make the best moves possible at every step of the game. To force a draw you need to follow a particular strategy, which isn't too hard, but tedious to write down (see here).

But now imagine you play tic-tac-toe the other way around, so that the first person to get three in a row loses the game. Again the game will end in a draw if both players play optimally. But this time the strategy for the first player is really easy: at your first move, choose the central square, and after that simply mirror your opponent's moves. By "mirror" we mean that you choose the square that is diametrically opposite the last square the opponent chose: if the opponent chose the top right corner, you choose the bottom left one, if they chose the middle top one, you choose the middle bottom one, and so on.

The same works in a version of tic-tac-toe that has, not 3x3, but nxn squares, where n is an odd number (if n is even there is no central square). Again we play the version in which the player who first gets n in a row loses. If the first player adopts the strategy above, they will never lose.

It's not too difficult to see why this strategy works. Let's call the player who goes first A and the other one B. After A takes the central square at the first move, B goes next and A mirrors B from then on. This means that whatever configuration of squares A has taken, B has taken the mirror image of the same configuration first. In particular, if A was forced to take n squares in a row, then B must have taken n squares in a row at the previous move, so B would have lost before A even got to make the supposed losing move. Nice!

About this article

This article was inspired by content on our sister site Wild Maths, which encourages students to explore maths beyond the classroom and designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

September 29, 2014

Can we always find order in systems that are disordered? If so, just how large does a system have to be to contain a certain amount of order? In this video Imre Leader of the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge gives an equation free introduction to a fascinating area of maths called Ramsey theory.

This talk was originally given to an audience of Year 12 A-level maths students (aged 16-17) in June this year and formed part of a mathematics enrichment day organised by the Millennium Mathematics Project with a special focus on encouraging the development of mathematical thinking.

Intrigued? You can read a brief description of Ramsey theory or find out more in our more detailed article. And you can also find out how Ramsey theory gives us one of our favourite numbers.

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September 29, 2014

"This talk will mostly be stories ... I want to tell you about Albert Einstein, and about his theory of relativity — what it is, why he was thinking about it and also about some of the very latest developments that have happened just this year."

This video contains a talk by one of our favourite physicists, David Tong at the Department of Applied Mathematics and Theoretical Physics at Cambridge. It was originally given to an audience of Year 12 A-level maths students (aged 16-17) in June this year and formed part of a mathematics enrichment day organised by the Millennium Mathematics Project with a special focus on encouraging the development of mathematical thinking.

You can read more about Einstein and relativity on Plus!

September 23, 2014
Travel time map

You might be surprised at what our image of the week depicts – commuting times! It shows the areas accessible by public transport from Cambridge, leaving at 9.00am on a weekday morning. The coloured circles depict the travel time in bands of 10 minutes. Thank you to those kind souls travelling for over an hour for giving us so many beautiful colours! You can find out more about the image in our article Travel time maps – transforming our view of transport.

The picture is one of the images that appear in the book 50 visions of mathematics, which celebrates the 50th anniversary of the Institute of Mathematics and its Applications.

You can see previous images of the week here.

September 16, 2014

Mr and Mrs Huntrodd were both born on the same day — 19 September 1600. That's quite a coincidence isn't it? (See here just how much of a coincidence this is.) What's even more of a coincidence is that they both died, within 5 hours of each other, on the same day too, which was… drum roll please… 19 September 1680! This seems very surprising. So surprising that even the very hard to surprise Professor Sir David Spiegelhalter OBE, Winton Professor of the Public Understanding of Risk and friend of Plus, was surprised. So surprised that he took this photo and is now throwing a party to celebrate this unlikely event with party, this Friday 19 September.


(Image by David Spiegelhalter, image licensed under Creative Commons)

You can join Spiegelhalter and his risky colleagues Timandra Harkness and Michael Blastland at their Huntrodd's party or you can throw your own. Visit the website to find out more! Happy Huntrodd's day!

You can read more about coincidence in Spiegelhalter's Plus column or the Understanding Uncertainty site.

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September 11, 2014

We have several images of the week this week! And they all come from the beautiful design for the new maths gallery that will open at the Science Museum in London in 2016. The first is the inspiration that Zaha Hadid Architects used for the layout of the gallery. It's the turbulence field of an aircraft which captures how the air moves around it. This is mathematically described by the Navier Stokes equations, which apply in many different situations, from modelling the weather to understanding how blood flows. (You can read more about the Navier Stokes equations in Universal pictures and turbulence in Understanding turbulence.)


A top view of the turbulence field around the Handley Page aircraft, one of the main exhibits in the new maths gallery (Image courtesy of Zaha Hadid Architects)

Handley Page Gugnunc

The Handley Page Gugnunc in flight

The plane that will hang in the centre of the gallery is a Handley Page Gugnuncs. This experimental plane was designed for the Guggenheim Safe Aircraft competition in 1929 which aimed to find planes that could fly safely at slow speeds and with short takeoffs. The research beind the plane profoundly advanced aviation technology and understanding at the very beginnings of civilian air travel. (You can read two brilliant articles from the 1930s about the competition and the aircraft itself.)

Behind the plane are three minimal surfaces (surfaces with the smallest possible area that satisfy some constraints) that are based on the shapes of the vortices in the turbulence created behind the plane in flight. The equation defining these
  \[  \begin{array}{l} a_1 sin(x)sin(2y)sin(3z) \\ + a_2 sin(2x)sin(y)sin(3z)\\ +a_3 sin(2x)sin(3y)sin(z)\\ +a_4 sin(3x)sin(y)sin(2z)\\ +a_5 in(x)sin(3y)sin(2z)\\ +a_6sin(3x)sin(2y)sin(z)=0 \end{array}  \]    
The plane exhibit

The design or the Handley Page exhibit in the new maths gallery

is a complicated looking combination of familiar sine functions that is governed by six different parameters. By varying the values of these parameters, the $a_ i$, the surface bends and twists producing a whole family of different shapes. Some of these will be created will be created in physical form to provide the support display cases used throughout the galleries. You can play with these surfaces yourself or just watch their creation in the video below.

It is thrilling to see such a striking, beautiful and mathematical design for the gallery. But of course the real strength will be the exhibits themselves. These will span 400 years of science and mathematics with objects ranging from hand-held mathematical instruments from the renaissance to modern technology. We all know that maths is the language of the universe and we can't wait to see what stories we'll find in the new maths gallery when it opens in 2016.

You can find out more about the gallery at the Science Museum

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