Science

Can you solve it? The enduring appeal of Venn diagrams


John Venn 1834-1923



John Venn 1834-1923

John Venn – the British logician who around 1880 devised the ‘Venn diagram’ – celebrates his 185th birthday this week.

His idea, a brilliantly simple visual aid to understanding logical relations, is one of the few concepts from abstract mathematics that is easily understood, and regularly used, by non-mathematicians.

Indeed, only last week it was revealed that Boris Johnson had planned his Cabinet by drawing a Venn diagram: in one circle he put Brexiteers, and, in the other, people capable of running the country. (It was not revealed, however, whether the number in the intersection reached double, or even single, figures.)

Because they need very little explaining, and involve straightforward logical deduction, Venn diagrams make for lovely puzzles. Here’s four of them:

The first three are by superstar maths teacher, author and podcaster Craig Barton.

1) Think of a fraction that could belong in each of the regions marked A to D below, or say that it is impossible. (Each circle represents the set of fractions described by its rule.)

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2) Think of a number that could belong in each of the regions marked A to H below, or say that it is impossible. (A factor of a number is a number that divides into that number with no remainder.)

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3) Think of 5 numbers that that could belong in each of the regions marked A to H below, or say that it is impossible. (The mean is the sum of the numbers divided by 5, the median is the middle number when listed in order of size, the mode is the most common number and the range is the difference between the highest and the lowest number.)

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Before we get to the final question, a humorous interlude. Designer Stephen Wildish has made a name for his witty Venns, of which here are two favourites.

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Illustration: Stephen Wildish

Mr Misters



Mr Misters Photograph: Stephen Wildish

Astute observers will have noticed that the above image is not strictly a Venn diagram, since not all relations between all the sets are possible. For example, where is the region inside the sets ‘strong’ and ‘rich’, but outside all the other sets? It doesn’t exist.

The message here is that while circles are fine when drawing Venn diagrams with two and three sets, they don’t work for four sets and more. Which brings us to the final question:

4. Draw a four set Venn diagram. The diagram needs to include four sets and all possible regions shared between the sets, i.e. there is a region in any one set, and outside all others; there is a region in any two sets and outside the others; there is a region in any three sets and outside the other; there is a region inside all sets.

I’ll be back with the answers at 5pm UK.

PLEASE NO SPOILERS

If you want to tweet or email me Venn diagrams with the answers to any of today’s questions, please do. I’ll include the best ones when I post the answers later today. In fact, if you can devise any funny Venn diagrams – in the manner of Stephen Wildish (or Boris Johnson) – send them in too.

Thanks to Craig Barton. His website Maths Venns contains many problems like the ones above. You can also find a huge amount of free material for teachers and students at Mr Barton Maths. He also has a podcast, and has written a critically-acclaimed book How I Wish I’d Taught Maths.

Thanks to Stephen Wildish of Wildish and Co. Many of his best images are included in his book Chartography.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

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I’m the author of Football School, a book series for 7 to 12 year olds that opens up the curriculum through football. Football School Star Players: 50 Inspiring Stories of True Football Heroes is just out. It profiles 50 footballers – including Women’s World Cup heroes Lucy Bronze and Megan Rapinoe – who show that football can be a force for good both on and off the pitch.





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