Tag Archives: calculus

Hjelmslev’s Theorem

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.

See Hjelmslev’s Theorem

Pythagorean Parabola Puzzle

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
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The Tired Messenger Problem

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

Fireworks Rocket

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.

Minimum Path Via Circle

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”.

See Minimum Path Via Circle

A Divine Language

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

Wine Into Water Problem

Here is a challenging problem from the 1874 The Analyst.

“A cask containing a gallons of wine stands on another containing a gallons of water; they are connected by a pipe through which, when open, the wine can escape into the lower cask at the rate of c gallons per minute, and through a pipe in the lower cask the mixture can escape at the same rate; also, water can be let in through a pipe on the top of the upper cask at a like rate. If all the pipes be opened at the same instant, how much wine will be in the lower cask at the end of t minutes, supposing the fluids to mingle perfectly?

—  Communicated by Artemas Martin, Mathematical Editor of Schoolday Magazine, Erie, Pennsylvania.”

I found the problem in Benjamin Wardhaugh’s book where he describes The Analyst:

“Beginning in 1874 and continuing as Annals of Mathematics from 1884 onward, The Analyst appeared monthly, published in Des Moines, Iowa, and was intended as “a suitable medium of communication between a large class of investigators and students in science, comprising the various grades from the students in our high schools and colleges to the college professor.” It carried a range of mathematical articles, both pure and applied, and a regular series of mathematical problems of  varying difficulty: on the whole they seem harder than those in The Ladies’ Diary and possibly easier than the Mathematical Challenges in the extract after the next. Those given here appeared in the very first issue.”

I tailored my solution after the “Diluted Wine Puzzle”, though this problem was more complicated.  Moreover, the final solution must pass from discreet steps to continuous ones.

There is a bonus problem in a later issue:

“19.  Referring to Question 4, (No. 1): At what time will the lower cask contain the greatest quantity of wine?

—Communicated by Prof. Geo. R. Perkins.”


See the Wine Into Water Problem for solutions.