Plus Blog

December 16, 2013

Here's a dilemma. Suppose you and a friend have been arrested for a crime and you're being interviewed separately. The police offer each of you the same deal. You can either confess, incriminating your partner, or remain silent. If you confess and your partner doesn't, then you get 2 years in jail (as a reward for talking), while your partner gets 10 years. If you both confess, then you both get 8 years (reduced from 10 years because at least you talked). If you both remain silent, you both get 5 years, as the evidence is only sufficient to convict you of a lesser crime.

Solitary confinement

Solitary confinement

What should your strategy be? As a selfish and rational individual, you should talk. If your partner also talks, then your confession gets you 8 years instead of 10. If your partner doesn't talk, then it gets you 2 years instead of 5. Talking is your dominant strategy, it leaves you better off than silence, no matter what your partner does.

The trouble is that your partner, just as selfish and rational as you, will come to the same conclusion. You'll both decide to talk and get 8 years each. Paradoxically, your dominant strategy will leave both of you worse off than silence would have done.

The prisoner's dilemma is one of game theory's most famous games because it illustrates why people might refuse to cooperate when they would be better off doing so. One real-life situation that is similar to the dilemma is an arms race between two countries, in which both countries increase their military might when it would be better for both to disarm.

The dilemma has been used extensively in mathematical research into altruism. Mathematical research into altruism? Yes, that's right! Using the dilemma as the basis for computer simulations in which simulated individuals can either cooperate or defect has shown how altruism can evolve as a survival strategy, even in large societies.

To find out more, read Does it pay to be nice? And there's more about the prisoner's dilemma and economics in Adam Smith and the invisible hand.

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December 15, 2013
Lever

Image © 2004 César Rincón.

Kneeling in the mud by a country road on a cold drizzly day recently, we finally appreciated the wonder that is a lever. We were trying to change a flat tyre and even jumping on the end of the wheel wrench wouldn't budge the wheel nuts. But when the AA arrived they undid them with ease, thanks to a wheel wrench that was three times the size of ours. There you have it ... size really does matter!

A lever is a truly remarkable device that can literally give any of us the strength of ten men. You can counteract 10 men pushing down on one side of a see-saw by applying just 1/10th of their force, as long as you are 10 times further from the see-saw's centre as they are. This is because the forces acting on a lever are proportional to the distances they are from the fulcrum. In this way a small amount of force moving a longer distance can move a large load over a smaller distance.

Levers are working hard all around us: in see-saws (where the fulcrum is between the loads), in wheel barrows (where the load is between the fulcrum and the force) and even in our very jaws (where the force is applied between the fulcrum and the load).

Archimedes was the first to mathematically describe how levers work and famously said: "Give me fulcrum and I shall move the Earth with a lever." And give us a long enough wheel wrench and we might just be able to change my next flat tyre for ourselves!

You can read more about mechanics on Plus.

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December 14, 2013

A polyhedron is the 3D version of a polygon. It's a solid object whose surface is made up of a number of polygonal faces. Two faces meet in an edge and the corners of a polyhedron are called vertices.

Euler's polyhedron formula, named after Leonhard Euler, is a pretty amazing equation relating the number $E$ of edges, $V$ of vertices and $F$ of faces of a polyhedron:

  \[  V-E+F=2  \]    
As an example, think of a cube. It's got 6 faces, 8 vertices and 12 edges:
  \[  8-12+6=2  \]    
as required.

The amazing thing is that this formula holds for all polyhedra, except for those that have holes running through them.

Using this formula, you can figure out quickly that there is no simple polyhedron (that is one without holes) with exactly seven edges. Similarly, there is no simple polyhedron with ten faces and seventeen vertices.

The formula also makes it possible to prove one of the most beautiful results in geometry, that there are only five Platonic solids.

The Platonic solids

Figure 7: The Platonic solids. From left to right we have the tetrahedon with four faces, the cube with six faces, the octahedron with eight faces, the dodecahedron with twelve faces, and the icosahedron with twenty faces.

Euler's polyhedron formula applies to solids that, in a topological sense, are equivalent to the sphere: you can turn each simple polyhedron into a sphere by smoothing out the edges and corners and making it round. But you can also look at the number $V-E+F$ for polyhedra that are topologically the same as a doughnut, or a doughnut with two, three, four or any number of holes. This number is called the Euler characteristic and it turns out that it is always equal to 2 minus 2 times the number of holes. So the Euler characteristic doesn't only tell us something about polyhedra, but also something very deep about the nature of shape and space in general.

You can find out more about Euler's polyhedron formula, including a proof, in this Plus article. And there's an interesting application to designing footballs in A fly walks around a football.

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December 13, 2013

No, this isn't about Homer, but it's interesting anyway. Suppose you're trying to decide which university to go to. You find out that last year the university you're interested in admitted 30% of male applicants but only 21.3% of female applicants. Looks like a clear case of gender bias, so you're tempted to go somewhere else. But then you look at the figures again, this time broken up by department. The university only has two departments, maths and English. The English department admitted 40% of male applicants and 42% of female applicants. The maths department admitted 10% of male applicants and 11% of female applicants. So if you look at the figures by department, if anything there's bias in favour of women. What's going on?

Students

This is an example of Simpson's paradox, which arises when you look at percentages without giving the actual numbers involved. Suppose the English department admits quite a high proportion of applicants, while the maths department is more choosy and admits only a small proportion. Now suppose that most of the male applicants apply to the English department. Then this drives up the overall percentage of successful male applicants, as English is easier to get into. Similarly, if most women apply to the maths department, then this lowers the overall percentage of successful women applicants, because maths is harder to get into. So it can happen that, although both departments favour women, the overall percentage of successful female applicants is lower than that for males.

Let's go back to the example: suppose that 100 men apply to the English department, so that means that 40 of them got in (40%). Suppose that only 50 women applied to the English department, so 21 of them got in (42%). Suppose the maths department had only 50 male applicants, so 5 got in (10%), and 100 female applicants, of which 11 got in (11%). Then the overall proportion of male applicants who were successful is 45/150 corresponding to 30%. For the women the overall proportion is 32/150 corresponding to 21.3%. Mystery solved.

This paradox isn't just a theoretical curiosity. In 1973 The University of California at Berkeley was sued for sex bias on the basis of figures that were an illustration of Simpson's paradox. It turned out that on the whole women had applied to more competitive departments and that's how the seemingly biased figures arose.

You can find out more about Simpson's paradox on Plus.

If it's the other Simpsons you're interested in have a look at Simon Singh's book.

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December 12, 2013

There's nothing simpler than 1,2,3, ... we understand these numbers instinctively and that's why they're called the natural numbers. But if you really think about it, what are these numbers? How would you describe them to an alien devoid of a number instinct? Here's one way of defining them, developed by the Italian mathematician Giuseppe Peano:

aliens

Explaining numbers to aliens.

  1. First you proclaim that 1 is a natural number
  2. Then you say that every natural number n has a successor s(n), which you can also write as n+1.
  3. We also insist that this successor is never equal to 1
  4. And that different numbers have different successors.

These four rules give you all the natural numbers, neatly ordered in a line, starting from 1 (you could also have started from 0). They also give you arithmetic, since addition and multiplication are about repeatedly adding 1s and you know how to do this: you simply move up to the successor of the number you're looking at. Subtraction and division are just the reverse of addition and multiplication. So equipped with these rules your innumerate alien could actually do some pretty decent number theory.

The four rules form the basis of what's called Peano arithmetic. It's a formal mathematical system based on a set of axioms (which includes these four rules) together with a language in which to speak about numbers and rules for logical inference. In the beginning of the 20th century mathematicians hoped they could turn all of maths into one giant formal system similar to Peano's arithmetic. That way they could prove everything directly from the axioms, without any hidden assumptions, and make sure that maths contains no contradictions. But their dream was shattered in the 1930s by the logician Kurt Gödel, who showed that there are logical limits to what you can do using formal systems. Find out more in

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December 11, 2013

Lately we've been featuring a range of Sudoku-esque puzzles kindly provided by Muddled Puzzles. The puzzles owe their origin to Sudoku and go by the umbrella brand name Sudokion. Here is our latest addition: the Cross HyperLogikion. It's a challenging and beautiful specialty puzzle, only available in the Platinum (highest) challenge band. For more (and also less challenging) Sudokion puzzles, see our Sudokion page.

To see the rules of Cross HyperLogikion click on the "i" icon below.

Have fun!

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