Archive for the ‘Mathematics’ Category

Nepomuk goes for a ride

Tuesday, December 20th, 2011

I have a number of mathematical puzzles about a (ficticious) French friend of mine called Nepomuk. Here’s the first of a series:

Nepomuk is driving to Paris, and he passes a road sign telling him the distance left. Unluckily, he’s a bit short-sighted, and so all he sees on the sign is that the number of miles to Paris has three digits, with the middle one being a zero:

Nepomuk travels along at a constant speed, and after exactly one hour he passes a road sign telling him that he’s still YX miles from Paris — the same two digits appear, but inverted.

And yet another hour later (he’s still driving at the same constant speed), he sees a sign telling him that he’s only XY miles from Paris — the same digits as on the first sign, but without the zero.

What is the (constant) speed that Nepomuk is travelling at?

The elliptical trainer

Monday, December 5th, 2011

An “elliptical trainer” is an exercise machine, intended to give its user the same physical experience that he could get from climbing up stairs or just going for a walk. Here’s my “artist’s impression” of one of these things:

The basic idea is that the athlete (blue in this picture) steps up and down on two boards (only one shown here) which are attached to a wheel behind him (that can be slowed down with a brake to make him exert himself more) and to vertical bars that he holds on to, which swivel fore and back around a pivot marked “T” in this sketch.

All this time, the athlete’s foot remains on a spot “F” on the moving board, and apparently this point F moves along an elliptical curve (green in the sketch above) — hence the name of this machine.

This contraption was recently the subject of a limerick in the OEDILF, and in a private discussion with my fellow OEDILF author PGS (Holger Martin), the question arose whether the green curve in this sketch really is an ellipse, or only an egg-shaped non-circular shape. In other words: is the elliptical trainer really “elliptical”?

Obviously, this is just a simple question about planar geometry involving a circle, a circular arc and some straight lines, so I thought I’d be able to determine the green shape quite easily. It’s turned out to be a real pest though. I hope I can get it worked out later this week.

Correlation

Thursday, December 1st, 2011

I’ve been a member of the OEDILF, the Omnificent English Dictionary In Limerick Form, since February 2007, an online project attempting to define every word in the English language by a suitable limerick. When a limerick alone isn’t enough, one may add an “author’s note” to provide additional information. Over the last years, I’ve written a number of limericks defining mathematical terms — a subject shunned by most and feared by some other OEDILF authors, and I’ll be presenting some of them here. Here’s an old one from February 2008:

Correlation

Recent tests show that so-called “IQs”
Bear no strong correlation to who’s
Rich or poor, black or white,
Male or female, but might
Correspond with the size of your shoes.

Statistics do, in fact, prove a strong correlation between IQ-test results and shoe sizes. Young children tend to have very small feet, and their test results, given the same tests, hardly ever beat those of adults (whose feet are larger). Such cases show that correlation between two seemingly unrelated numbers (test results and shoe sizes) may be due to a hidden factor (age).

Many statistics of this kind seem profound and meaningful until the hidden connexion is revealed, and then they look trivial. Say someone claimed that a study found a large majority of all traffic accidents in Israel were caused by Jews. That might lead you to think that Jews are bad drivers — until you start asking who lives in Israel and might own a car. So don’t trust statistics alone; use your brain!