The photos to my recent post on the circumference remind me of another bet about a beer glass. Just ask your fellow drinkers the following question:

There are two glasses, one filled with beer and the other one with the same quantity of wine. One takes a spoonful of the beer and pours it into the wine; then one makes the the same in the other direction, i.e., one transfers a spoonful of that new wine-beer mix back into the glass filled with beer. Now in the first glass the beer is spoiled by a small quantity of wine and in the second glass the wine is spoiled by a small quantity of beer. Which glass is more polluted?

In my kitchen I have this cylindrical recipient for salt. Have a guess: what is longer its circumference or its height?

At first sight you would say, it is the height. Let's compare! My hand stretches easily to the height of the recipient but is not big enough to wrap my fingers around it. Quite surprising: the circumference is longer than the height!

We all learned in school, how to compute the circumference of a circle: you multiply the diameter by that famous number , which mathematicians invented just for that purpose and whose approximate value is 22/7. And since 22/7 is greater than 3, the circumference is bigger than three times the diameter. Keeping that in mind our result above is not so suprising any more. In fact, the height really looks like less than three times the width.

Of course, most people underestimate the length of the circumference. Next time in a pub, challenge your friends with that question. Bet a beer on it, then use a napkin, for example, to compare both lengths as I do in the pictures below... I am sure you will win your bet!

The height is smaller...	...than the circumference.	Mathoman wins a beer!

I am collecting interesting maths exercises that can be solved by undergraduate students. One can find a lot of them in textbooks, on the internet, in old worksheets of one's own student time... and sometimes one is lucky and invents a new exercise. Here is a linear algebra question that came up in my mind last week end. I like it because the solution I found needs no deep theorem but only some basic understanding of linear algebra:

What is the biggest integer k, such that every affine sub-space of codimension k in the space of n x n matrices contains an invertible matrix?

Recall: The codimension of a sub-space is the difference between the dimension of the ambient space and the one of the sub-space. In other words, it is the number of equations necessary to describe the subspace (every equation taking away one degree of liberty). For example, in our common three-dimenssional space the codimension of a line is 2 and the codimension of a plane is 1.