Computer pioneer, mathematician, educator, mentor, and friend Wally Feurzeig passed away on January 4. Wally was a key member of the small team that created the LOGO educational programming languages in the 60s. I' was privileged to know him and his lovely wife Nanni for over 15 years. Wally would listen patiently anc carefully to my various enthusiasms about math education, and always find something incredibly helpful to say, or introduce me to others from his circle such as Victor Gutenmacher and the late Oliver Selfridge who really expanded my understanding of what mathematics education could be. My wife Leslie and I were grateful for invitations to a Monday evening dinner that was cooked by his lovely wife Nanni, and featured some of their fascinating friends.
You can read about some of Wally's accomplishments on Wikipedia, but I just wanted to record a few personal remembrances here.
I am feeling the loss of a kind, gentle, and most intelligent man. We will not see Wally's like again.
The Four Kinds of Students
Richard Feynman is generally regarded as a great teacher, and I'd agree, based on my experience when I had him in sophomore physics at Caltech. However, he sometimes despaired of the teaching enterprise. He said something to the effect that teaching a concept is either unsuccessful (in the case of a poor student) or unnecessary (in the case of a good student who can pick it up by reading). I'd like to suggest that there are not just two, but four kinds of students.
The poor student is unable or unwilling to learn the material.
The mediocre student will learn as much as the teacher presents, but no more.
The good student will continue to learn after leaving the teacher.
The excellent student will surpass his or her teacher.
The poor student is unable or unwilling to learn the material.
The mediocre student will learn as much as the teacher presents, but no more.
The good student will continue to learn after leaving the teacher.
The excellent student will surpass his or her teacher.
A Sangaku Problem
I will be giving a talk Saturday at the NES-MAA meeting in Bridgewater MA in which I will talk about sangaku, or Japanese temple geometry problems. These problems, created by people from a wide walk of life, were beautifully drawn on wooden tablets which were then placed in Buddhist temples or Shinto shrines. Hundreds have been discovered, and probably thousands existed at one time. I became involved in this when I was challenged to solve one of the difficult sangaku. Here I present one of the easier ones, from the delightful book Sacred Geometry: Japanese Temple Geometry by Fukagawa Hidetoshi and Tony Rothman.
I chose this problem because the diagram is so beautiful, the solution is fairly simple, yet satisfying, and it is one of the few sangaku created by a woman (Okuda Tsume).
In a circle of diameter AB = 2R, draw two arcs of radius R with centers A and B respectively, and 10 inscribed circles, two green circles of diameter R, four red circles of radius t, and four blue circles of radius t'. Show that t = t' = R/6.
In the diagram below, we follow the convention of labeling the center of a circle with the radius of that circle.
I chose this problem because the diagram is so beautiful, the solution is fairly simple, yet satisfying, and it is one of the few sangaku created by a woman (Okuda Tsume).
In a circle of diameter AB = 2R, draw two arcs of radius R with centers A and B respectively, and 10 inscribed circles, two green circles of diameter R, four red circles of radius t, and four blue circles of radius t'. Show that t = t' = R/6.
In the diagram below, we follow the convention of labeling the center of a circle with the radius of that circle.
E22. A simple probability problem
How many rolls of one fair die does it take, on average, before all six numbers show up? Make a guess and then see if you can figure it out.
A6. Counting Triangulations
Here is a counting problem that was solved a long time ago. Feel free to try your hand at it.
Given P, a convex n-gon, a triangulation of P is a subdivision of P into n - 2 non-overlapping triangles. A triangulation is obtained by drawing n - 3 non-intersecting diagonals. Let f(n) be the number of different triangulations. Clearly, f(3) = 1, f(4) = 2, and f(5) = 5. Careful counting shows f(6) = 14. Find an expression for f(n).
Given P, a convex n-gon, a triangulation of P is a subdivision of P into n - 2 non-overlapping triangles. A triangulation is obtained by drawing n - 3 non-intersecting diagonals. Let f(n) be the number of different triangulations. Clearly, f(3) = 1, f(4) = 2, and f(5) = 5. Careful counting shows f(6) = 14. Find an expression for f(n).
The Waitress and the Mathematicians
I recently heard two stories on a LinkedIn math forum, under the topic of humor in mathematics. The first is a funny little story, which I first heard years ago. The second is a cute logic puzzle.
Story
Two mathematicians, Tom and Joe, are in a restaurant, discussing the state of mathematical illiteracy in the general public. Tom goes to the restroom, and Joe calls over the waitress and says, "I'd like to play a trick on my friend. I'll call you over and ask you a question. I'll give you ten dollars if you answer my question with 'x squared'". She agrees, and takes the money. Tom returns, and sometime later Joe says to Tom, "I'll bet most people know how to find the antiderivative of a simple function". Tom disagrees strongly, and Joe says. "OK, I'll bet that our waitress knows the antiderivative of 2x. If I'm wrong, I'll pay for lunch. If I'm right, you pay." Tom says "You're on."
The waitress comes over, and Joe asks her, "Excuse me, miss, but do you happen to know the antiderivative of 2x." The waitress replies "Sure. It's x squared ... plus C".
Puzzle
Four mathematicians come into a restaurant together, and a waitress comes over, and asks "Would you all like coffee?". The first mathematician says, "I don't know". The second mathematician says "I don't know". The third mathematician says "I don't know". The fourth mathematician says "no".
The waitress, who is no slouch at logic, comes back with the correct number of coffees. How many coffees did she bring?
Story
Two mathematicians, Tom and Joe, are in a restaurant, discussing the state of mathematical illiteracy in the general public. Tom goes to the restroom, and Joe calls over the waitress and says, "I'd like to play a trick on my friend. I'll call you over and ask you a question. I'll give you ten dollars if you answer my question with 'x squared'". She agrees, and takes the money. Tom returns, and sometime later Joe says to Tom, "I'll bet most people know how to find the antiderivative of a simple function". Tom disagrees strongly, and Joe says. "OK, I'll bet that our waitress knows the antiderivative of 2x. If I'm wrong, I'll pay for lunch. If I'm right, you pay." Tom says "You're on."
The waitress comes over, and Joe asks her, "Excuse me, miss, but do you happen to know the antiderivative of 2x." The waitress replies "Sure. It's x squared ... plus C".
Puzzle
Four mathematicians come into a restaurant together, and a waitress comes over, and asks "Would you all like coffee?". The first mathematician says, "I don't know". The second mathematician says "I don't know". The third mathematician says "I don't know". The fourth mathematician says "no".
The waitress, who is no slouch at logic, comes back with the correct number of coffees. How many coffees did she bring?
E21. Equiangular and Equilateral Polygons
A polygon is equiangular if all of its angles are equal. In particular, if the polygon has n sides, each angle measures (n - 2) * 180 / n degrees. A polygon is equilateral if all of its sides have the same length. It can be shown very easily that every equiangular triangle is equilateral. Of course, it is not true that every equiangular quadrilateral is equilateral. Any rectangle that is not a square provides a counterexample. Show that for every n > 3 there exists an equiangular n-gon that is not equilateral.
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