E12. Ladder against a wall (Part II)

Here is another ladder against a wall problem, from Coxeter's classic Introduction to Geometry. It is somewhat atypical of the book in that the interesting part seems to be the algebra, rather than the geometry. To make this more of a challenge, try to do it without using a CAS.

A 24-foot long ladder rests against the horizontal ground and a vertical wall in such a way that it touches a cube. The cube is 7 feet on a side and is placed flat on the ground, touching the wall. Find the height of the top of the ladder.

This is a two-dimensional problem which could be stated a little less colorfully in terms of a square and a line segment. It's easy to see that there must be (at least) two answers because of the symmetry of the problem; if a line segment of length 24 passes through (7,7) with endpoints on the positive coordinate axes, the reflection of that line segment in the line y = x will satisfy the same conditions. I found several ways of setting up the problem, all of which result in a 4th degree equation. However, the solutions are quadratic irrationals. In one approach, the biquadratic factors into two quadratics with integer coefficients. In another, a somewhat obvious substitution does the trick.

E11. Ladder against a wall

A ladder is placed against a (vertical) wall and the bottom of the ladder is moved away along the (horizontal) ground. What is the shape of the curve traced by the midpoint of the ladder?

It is very easy to work out the answer to this problem, and I won't bother to do that here. If you haven't seen the problem before, test your intuition. Try to sketch what you think curve looks like before solving the problem. (In particular, is the curve concave up or concave down?) The first time I saw this, my intuition was wrong.

Blogger etiquette

I was recently going through my old blog postings, and I found a thoughtful and positive comment by "Sarah" dated September 16, 2008. The original post was about research that purports to show that students learn mathematics better from abstract models rather than concrete ones. In the comment, Sarah apologizes for responding so late. (I'm not sure when the original post was; sometime in Summer 2008.) This is probably why I missed it.

Although I didn't have too much to add to what she said, I wanted to at least acknowledge her comment. I was able to visit her Blogger profile and her two blogs, but neither blog is terribly current and I would feel odd leaving my comment attached to a totally unrelated topic. However, if I were to leave the comment where logic dictates--on this blog, next to the original comment--it seems clear she would never see it.

I'd appreciate suggestions as to how to deal with this type of situation.

Lure of the Labyrinth

A couple of weeks back, I went to a meeting of the Association of Teachers of Mathematics in Massachusetts. The keynote speech was by Scot Osterweil of MIT's Educational Arcade. The goal is to produce games that teach mathematics in a way that is engaging to students and has true educational value. He showed us a game, Lure of the Labyrinth, that embodies these principles. It teaches mathematical topics, such as proportions, at the middle school level, and is quite engaging, even to adults. It reminded me a bit of Myst, although there is definitely a more kid-friendly feel.

Try it yourself by going to http://labyrinth.thinkport.org. You must register, but I don't think that there is any downside to that. You can choose either Game or Puzzles. I'd suggest trying Puzzles first, and selecting the first puzzle.

I really liked this and I agree with Scot that properly designed puzzles and games are one of the best ways to teach mathematics.

A New Business Card

I recently designed a new business card, using an interesting geometrical structure as a design element. The design is based on a circular Dirichlet tessellation, also known as a Voronoi diagram with multiplicative weights. The design seemed appropriate because I have done research on these structures in the past, in a paper I wrote with Ethan Bolker in the eighties.

In the multplicative Voronoi diagram, we start with a finite number of sources (points) in the plane, each assigned a positive weight w. The diagram consists of the circles or circular arcs that divide the plane into regions, where the region corresponding to point P consists of all points X such that
|P-X|/w(P) is less than or equal to |Q-X|/w(Q) for every other source Q.

You can think of the sources as being restaurant locations and the weights as being a desirability rating, so if w(P) is r times w(Q), a customer is willing to travel r times as far to go to P as to go to Q. For the case of two sources, the boundary is the circle of Apollonius of ratio r. The case where all weights are equal reduces to the classical Voronoi diagram, where the circular arcs degenerate into straight lines.

If you would like to play around with these diagrams, you can use the applet written by Gabi Knuppertz at http://www.pi6.fernuni-hagen.de/GeomLab/VoroMult/. I was not able to find the needed plugin for Firefox, but got it to work fine in Internet Explorer.

E10. A formula for weighted totals

Professor Blank wants to use a spreadsheet to compute for each student a weighted total of four quiz grades, where the students highest grade is multiplied by four, the next highest grade by three, the next highest by two, and the lowest by one. He requests a formula that will calculate this weighted total using only addition, subtraction, multiplication by an integer, and the min and max functions. Find such a formula.

The problem is a little harder than it appears at first. For a solution, go here.

Freeman Dyson's Problem

My friend John Lamperti turned my attention to a number theory problem in an article on Freeman Dyson in the March 29 New York Times Magazine Section. It is an excellent article, which I recommend. The section with the problem is the following:

[T]aking problems to Dyson is something of a parlor trick. A group of scientists will be sitting around the cafeteria, and one will idly wonder if there is an integer where, if you take its last digit and move it to the front, turning, say, 112 to 211, it’s possible to exactly double the value. Dyson will immediately say, “Oh, that’s not difficult,” allow two short beats to pass and then add, “but of course the smallest such number is 18 digits long.” When this happened one day at lunch, William Press remembers, “the table fell silent; nobody had the slightest idea how Freeman could have known such a fact or, even more terrifying, could have derived it in his head in about two seconds.” The meal then ended with men who tend to be described with words like “brilliant,” “Nobel” and “MacArthur” quietly retreating to their offices to work out what Dyson just knew.

The discovery (or proof) of the smallest such number, 105263157894736842, makes a good problem for an elementary number theory course or a bright high school student.

The first time John told me the problem, he had heard it second-hand, and it was backwards: Is there an integer which if you take its first digit and move it to the back you can exactly double the value? In this case the answer is no, and the proof is simpler than the solution for Dyson’s problem.

To see my solutions, go here.