This is another puzzle from BL’s Weekly Math Games.
“a + b + c = 2, and
a2 + b2 + c2 = 12
where a, b, and c are real numbers. What is the difference between the maximum and minimum possible values of c?”
The original problem statement mentioned a fourth real number d, but I considered it a typo, since it was not involved in the problem.
See Sphere and Plane Puzzle for a solution.
I came across this remarkable result in Futility Closet:
“On each of these two black lines is a trio of red points marked by the same distances. The midpoints of segments drawn between corresponding points are collinear.
(Discovered by Danish mathematician Johannes Hjelmslev.)”
This result seems amazing and mysterious. I wondered if I could think of a proof. I found a simple approach that did not use plane geometry. And suddenly, like a magic trick exposed, the result seemed obvious.
Well, I discovered that the 2024 Math Calendar has some interesting problems, so I guess things will limp along for a while. This is a challenging but imaginative problem from the calendar.
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As before, recall that all the answers are integer days of the month.
See the Amazing Root Problem for a solution.
This problem is from Colin Hughes’s Maths Challenge website (mathschallenge.net).
“Four corners measuring x by x are removed from a sheet of material that measures a by a to make a square based open-top box. Prove that the volume of the box is maximised iff the area of the base is equal to the area of the four sides.”
See the Maximized Box Problem
Since the changes in Twitter (now X), I have not been able to see the posts, not being a subscriber. But I noticed poking around that some twitter accounts were still viewable. However, like some demented aging octogenarian they had lost track of time, that is, instead of being sorted with the most recent post first, they showed a random scattering of posts from different times. So a current post could be right next to one several years ago. That is what I discovered with the now defunct MathsMonday site. I found a post from 10 May 2021 that I had not seen before, namely,
“The points A and B are on the curve y = x2 such that AOB is a right angle. What points A and B will give the smallest possible area for the triangle AOB?”
See the Pythagorean Parabola Puzzle for solution.
(Update 9/1/2023) Elegant Alternative Solution by Oscar Rojas
Continue reading
Here is another challenging problem from the Polish Mathematical Olympiads. Its generality will cause more thought than for a simpler, specific problem.
“A cyclist sets off from point O and rides with constant velocity v along a rectilinear highway. A messenger, who is at a distance a from point O and at a distance b from the highway, wants to deliver a letter to the cyclist. What is the minimum velocity with which the messenger should run in order to attain his objective?”
See the Tired Messenger Problem
This is another physics-based problem from Colin Hughes’s Maths Challenge website (mathschallenge.net) that may take a bit more thought.
“A firework rocket is fired vertically upwards with a constant acceleration of 4 m/s2 until the chemical fuel expires. Its ascent is then slowed by gravity until it reaches a maximum height of 138 metres.
Assuming no air resistance and taking g = 9.8 m/s2, how long does it take to reach its maximum height?”
I can never remember the formulas relating acceleration, velocity, and distance, so I always derive them via integration.
See the Fireworks Rocket for solutions.
James Tanton provides another imaginative problem on Twitter.
“I am at point A and want to walk to point B via some point, any point, P on the circle. What point P should I choose so that my journey A → P → B is as short as possible?”
Hint: I got ideas for a solution from two of my posts, “Square Root Minimum” and “Maximum Product”.
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?
Here is another sum problem, this time from the 2021 Math Calendar.
________________
As before, recall that all the answers are integer days of the month. And the solution employs a technique familiar to these pages.
See the Winter Sum for a solution.
