Here is a problem from the 2022 Math Calendar.
“Two small circles of radius 4 are inscribed in a large semicircle as shown. Find the radius of the large semicircle.”
As before, recall that all the answers are integer days of the month.
As seemed to be implied by the original Math Calendar diagram, I made explicit that the upper circle was tangent to the midpoint of the chord. Otherwise, the problem is insufficiently constrained.
See Two and a Half Circles for a solution.
Unfortunately Ukraine has receded from our attention under the threat from our own anti-democratic forces, but this Monday Puzzle from Alex Bellos in March is a timely reminder of the mathematical significance of that country.
“Like many of you I’ve hardly been able to think about anything else these past ten days apart from the war in Ukraine. So today’s puzzles are a celebration of Lviv, Ukraine’s western city, which played an important role in the history of 20th century mathematics. During the 1930s, a remarkable group of scholars came up with new ideas, methods and theorems that helped shape the subject for decades.
The Lwów school of mathematics – at that time, the city was in Poland – was a closely-knit circle of Polish mathematicians, including Stefan Banach, Stanisław Ulam and Hugo Steinhaus, who made important contributions to areas including set-theory, topology and analysis. …
Of the many ideas introduced by the Lwów school, one of the best known is the “ham sandwich theorem,” posed by Steinhaus and solved by Banach using a result of Ulam’s. It states that it is possible to slice a ham sandwich in two with a single slice that cuts each slice of bread and the ham into two equal sizes, whatever the size and positions of the bread and the ham.
Today’s puzzles are also about dividing food. The first is from Hugo Steinhaus’ One Hundred Problems in Elementary Mathematics, published in 1938. The second uses a method involved in the proof of the ham sandwich theorem.
Show there is always a diameter that cuts the circle into two batches of 12 rolls with an equal number of plain and seeded.
Question 2 is adapted from Mathematical Puzzles by Peter Winkler, who gives as a reference Alon and West, The Borsuk-Ulam Theorem and bisection of necklaces, Proceedings of the American Mathematical Society 98 (1986).”
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?
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 for solutions.
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:
We can conclude from these two sentences that:
Actually, the question is which, none or more, of statements (A) – (E) follow from the two sentences?
See Pinocchio’s Hats for solutions.
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.
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 for a solution.
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 for solutions.
