I have just finished reading a most remarkable book by Alec Wilkinson, called A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age. I had read an essay of his in the New Yorker that turned out to be essentially excerpts from the book. I was so impressed with his descriptions of mathematics and intrigued by the premise of a mature adult in his 60s revisiting the nightmare of his high school experience with mathematics that I was eager to see if the book was as good as the essay. It was, and more.
The book is difficult to categorize—it is not primarily a history of mathematics, as suggested by Amazon. But it is fascinating on several levels. There is the issue of a mature perspective revisiting a period of one’s youth; the challenges of teaching a novice mathematics, especially a novice who has a strong antagonism for the subject; and insights into why someone would want to learn a subject that can be of no “use” to them in life, especially their later years.
Wilkinson has a strong philosophical urge; he wanted to understand the role of mathematics in human knowledge and the perspective it brought to life. He was constantly asking the big questions: is mathematics discovered or invented, what is the balance between nature and nurture, why does mathematics seem to describe the world so well, what is the link between memorization and understanding, how do you come to understand anything?
See A Divine Language
Henk Reuling posted a deceptively simple-looking geometric problem on Twitter.
“I found this old one cleaning up my ‘downloads’ [source unknown] I haven’t been able to solve it, so help!
According to the given information in the figure, what is the length of the missing interval on the diagonal of the square?”
See the Missing Interval Puzzle.
This problem in logic from Presh Talwalkar recalled an article I wrote a while ago but did not publish. So I thought I would post it as part of the solution.
“Assume that both of the following sentences are true:
- Pinocchio always lies;
- Pinocchio says, “All my hats are green.”
We can conclude from these two sentences that:
- (A) Pinocchio has at least one hat.
- (B) Pinocchio has only one green hat.
- (C) Pinocchio has no hats.
- (D) Pinocchio has at least one green hat.
- (E) Pinocchio has no green hats.”
Actually, the question is which, none or more, of statements (A) – (E) follow from the two sentences?
See Pinocchio’s Hats
As I am sure is common with most mathematicians, I had become interested in the history of the development of mathematical symbols, first for numbers (numerals) and then for algebra (symbolic algebra). Joseph Mazur’s book Enlightening Symbols provided an excellent history of this evolution. His focus on the development and significance of symbolic algebra in the Renaissance was especially illuminating. I also augmented Mazur’s information with details from Albrecht Heeffer’s work.
Such a subject cries out for a timeline to appreciate the order and timing of discoveries, which Mazur provided, concentrating on the Renaissance. I decided to both simplify Mazur’s version and expand it to cover the evolution of numbers and their notation, as well as to set the whole enterprise in the context of historical periods.
See Symbolic Algebra Timelines
This math problem from Colin Hughes’s Maths Challenge website (mathschallenge.net) is a bit more challenging.
“In the diagram, AB represents the diameter, C lies on the circumference of the circle, and you are given that
(Area of Circle) / (Area of Triangle) = 2π.
Prove that the two smaller angles in the triangle are exactly 15° and 75° respectively.”
See the 15 Degree Triangle Puzzle
This work problem from Geoffrey Mott-Smith is a little bit tricky.
“An engineer working on the Alcan Highway was heard to say, “At the time I said I could finish this section in a week, I expected to get two more bulldozers for the job. If they had left me what machines I had, I’d have been only a day behind schedule. As it is, they’ve taken away all my machines but one, and I’ll be weeks behind schedule!”
How many weeks?”
See the Alcan Highway Problem
This is a nice puzzle from the Scottish Mathematical Council (SMC) Senior Mathematical Challenge of 2008. It is more a logic puzzle than a geometric one.
“In the diagram, each question mark represents one of six consecutive whole numbers. The sum of the numbers in the triangle is 39, the sum of those in the square is 46 and the sum of those in the circle is 85. What are the six numbers?”
See the Shared Spaces Puzzle
This is another infinite series from Presh Talwalkar, but with a twist.
“This problem is adapted from one given in an annual national math competition exam in France. Evaluate the infinite series:
1/2! + 2/3! + 3/4! + …”
The twist is that Talwalkar provides three solutions, illustrating three different techniques that I in fact have used before in series and sequence problems. But this time I actually found a simpler solution that avoids all these. You also need to remember what a factorial is: n! =n(n – 1)(n – 2)…3·2·1.
See a Nice Factorial Sum
This is another fairly simple puzzle from Futility Closet from a while ago (2014).
“Two lines divide this equilateral triangle into four sections. The shaded sections have the same area. What is the measure of the obtuse angle between the lines?”
See Sizing Up
This is a slightly challenging problem from the 1993 American Invitational Mathematics Exam (AIME).
“Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Find t, the amount of time in seconds, before Jenny and Kenny can see each other again.”
See the Parallel Stroll Problem