18 November 2008

Silly maths quiz at BBC site

Marcus du Sautoy, the man who will succeed Richard Dawkins as professor for the understanding of science at the University of Oxford, has a mission to break down people's fear of maths. So he set up a quiz. Unfortunately, the quiz is rather naive and does not reflect what mathematics is about. Here are some dubious questions:

* A temple was built in 50 BC and burnt down 75 years later. In what year whas it burnt down?
(1) 24 AD. (2) 25 AD. (3) 26 AD.
The logical thinking gives -50+75=+25, i.e. 25 AD (provided that both events took place on the same day of the year, say on day 1.)
But the answer given by the BBC is: 26 AD. There was no year 0.
This is a fact, indeed: that there was no year 0. Some 1500 years ago, a monk called Dionisius Exiguus was charged with the creation of a calendar; this monk did not know that the integer succeeding -1 is 0 and not +1; i.e. he thought that -1+1=1.
But the ignorance of a monk should not be brought into a mathematical quiz which, supposedly, tests one's understanding of logic (and not of history!)

* Imagine picking up a cube with your thumb placed at one corner of the cube and your finger at the furthest corner on the other side of the cube. Now cut the cube exactly down the middle between your finger and thumb. You have two pieces with a new face. What is the shape of the new face?
(1) Hexagonal. (2) Square. (3) Triangular.
Before looking at the answer let us look at the way the problem is phrased. First, what exactly is meant by a corner? Is it a vertex? Is it an edge? I think he means a vertex, i.e. the point at which 3 faces meet. But it could be an edge: the segment at which 2 faces meet forms a corner, in my humble opinion, too. The answer will of course, be different depending on the interpretation of the word corner. OK, let us assume he means a vertex. Now what does he mean by "cut the cube exactly down the middle"? This could also mean a number of things. Here we must guess that he means to take the plane that is perpendicular to the segment joining two vertices at farthest distant appart and which passes through the middlepoint of this segment. This plane intersects the (boundary of the) cube at a polygon. And this polygon is, indeed, a hexagon.
But a mathematical question should be phrased in an unambiguous way!

* What is the next number in the sequence 1001, 122, 101, 32, 25, 23, 21, ...
(1) 20. (2) 18. (3) 15.
Again, this is a dangerous question for the mathematically uninitiated for he or she may think that there is a unique choice. For instance, if I give you the sequence 1,4,8,16,32,64,128 you might think that the next number is 256 (powers of 2). But a perfectly acceptable answer is 144. (Indeed, if we let p(m) be the product of the exponents in the prime factorisation of m, and define q(n) to be to be the smallest integer such that p(q(n))=n then q(1)=1, q(2)=4,...,q(7)=128, but q(8)=144.)
Now then, the answer to this question, according to BBC, but also according to the encyclopedia of integer sequences, is 18: if we write the number 17 in base 2 we get 1001, in base 3 it is 122, and so on.
But hang on a minute! This is by far NOT a fair question and a misleading one for the two reasons I outlined: first, it gives the wrong impession on what Mathematics is about. Second, the answer is hard and, clearly, not unique.

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What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant