Plus Blog

December 2, 2015

Sometimes real progress in maths comes when you find a way of looking at a problem in two different ways. Here is a great example of this.

handshake

Suppose you have $n+1$ people in a room and each person shakes hands with each other person once. How many handshakes do you get in total? The first person shakes hands with $n$ other people, the second shakes hands with the $n-1$ remaining people, the third shakes hands with $n-2$ remaining people, etc, giving a total of

$n+(n-1)+(n-2)+...+2+1$ handshakes.

But we can also look at this in another way: each person shakes hands with $n$ others and there are $n+1$ people, giving $n \times (n+1)$ handshakes. But this counts every handshake twice, so we need to divide by 2, giving a total of

  \[ \frac{n \times (n+1)}{2} \]    

handshakes.

Putting these two arguments together, we have just come up with the formula for summing the first $n$ integers and we’ve proved that it is correct:

  \[ n+(n-1)+(n-2)+...+2+1 = \frac{n \times (n+1)}{2}. \]    

This puzzle is inspired by content on our sister site Wild Maths, which encourages students to explore maths beyond the classroom and is 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.

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December 1, 2015

Only three things in life are certain: death, taxes and parking fees. But even a menacing parking meter is an excuse to do some maths.

Mini

Imagine, for example, that the car park costs £1.50. The machine only accepts 10p and 20p coins. There are obviously different ways of putting the money into the car park machine, for example

10p, 10p, 20p, 20p, 10p, 10p, 10p, 10p, 20p, 10p, 10p, 10p

or

10p, 10p, 10p, 10p, 20p, 20p, 20p, 10p, 20p, 20p.

You could probably go for the rest of the month without feeding the machine in the same way twice. Can you feed the machine in a different way each day of the year?

You can find a longer version of this puzzle, including some follow-up questions to investigate, on the Wild Maths site. Wild Maths encourages students to explore maths beyond the classroom and is 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.

Return to the Plus Advent Calendar

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October 29, 2015

Today is national cat day in the US! To mark the occasion, here's a quick introduction to the most famous cat in the history of science: Schrödinger's cat.

Schrödinger's cat

Schrödinger's cat. Image: Dhatfield.

One interpretation of the strange theory of quantum mechanics is that tiny particles can simultaneously exist in states that we would usually deem mutually exclusive. For example, an electron can be in two places at once, or a radioactive atom can be both decayed an non-decayed at the same time. It's only when we go to measure a system in superposition, as this strange state is called, that reality somehow "collapses" to one of the possibilities.

In 1935 the physicist Erwin Schrödinger, who made major contributions to the theory of quantum mechanics, developed a thought experiment in order to demonstrate just how counter-intuitive the idea of superposition is. We let him describe it in his own words, taken from a translation of his 1935 paper:

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid.

Thus, when an atom decays, poison will be released from the flask and the cat killed. And here's the main point. If it is true that, as long as we don't look, the system can evolve into a superposition state of atoms being simultaneously decayed and not decayed, then it follows that, as long as we don't look, the cat will be simultaneously dead and alive. Poor cat. Or should we say lucky cat?

You can find out more in Schrödinger's equation — what is it? and Schrödinger's equation — what does it mean?.

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October 27, 2015
Book cover

Barrow has written a book about maths and the arts.

John D. Barrow, mathematician, cosmologist and boss of Plus, explores maths and the arts in a public talk in Cambridge on Monday, 02 November 2015. Barrow will look at ways in which maths can shed light upon a range of questions in the arts and how problems of art and design inspire new mathematical questions. The canvas will be broadly drawn with examples from different areas of the arts, including painting, textual analysis, diamond cutting, Henry Moore's stringed figures, ballet, and even the best place to stand when viewing statues.

The talk is from 19:30 to 21:00 at Churchill College, Wolfson Lecture Theatre, Storeys Way, Cambridge CB3 0DS. It is organised by the Cambridge Society for the Application of Research. Non-members will be asked to pay a nominal entry fee of £3.00. Find out more here.

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August 18, 2015

In this fascinating talk theoretical physicist Ben Allanach talks about the search for dark matter at the Large Hadron Collider, including a generous helping of information on the Higgs boson. Ben gave the talk on 19 June 2015 to an audience of school students aged 16-17 as part of a mathematics enrichment event at the University of Cambridge.

If you prefer read, the see the abridged version of this talk.

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August 4, 2015

In this excellent talk the mathematician Vicky Neale gives a fascinating and easy-to-follow introduction to the prime numbers — from a thorough description of what they are, via the ancient proof that there are infinitely many, to the prime number theorem, the twin prime conjecture and more. By the end of this talk you hopefully agree with us, and Vicky, that the world would be a very boring place without primes.


Vicky gave this talk at the University of Cambridge on 19 June 2015 to an audience of Year 12 A-level Maths students (aged 16-17). It formed part of a mathematics enrichment day organised by the Millennium Mathematics Project with a special focus on encouraging creative mathematical thinking. For more on creative mathematics, visit Wild Maths.

To find out more about prime numbers, browse our articles on the subject.

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