This problem comes from the defunct Wall Street Journal Varsity Math Week collection.
“The coach then shows the team the diagram to the left and asks: What is the maximum area of a rectangle contained entirely within a triangle with sides of 9, 10 and 17?”
I changed the numbers a bit to make my calculations easier, but left the problem otherwise unchanged. When I checked the Varsity Math Week solution, I saw they used a simplifying formula that I could not remember. I also believed their solution left out a justification for the maximal area. Besides an intuitive solution for this, I also included a calculus version. See the Triangular Boundary Problem.
Yet another Futility Closet puzzle.
“Point E lies on segment AB, and point C lies on segment FG. The area of parallelogram ABCD is 20 square units. What’s the area of parallelogram EFGD?”
I had an alternative solution that I thought was a bit simpler and clearer. See the Parallelogram Puzzle.
This is another Futility Closet puzzle.
“Four straight roads cross a plain. No two are parallel, and no three meet in a point. On each road is a traveler who moves at some constant speed. If Blue and Red meet each other at their crossroad, and each of them meets Yellow and Green at their respective crossroads, will Yellow and Green necessarily meet at their own crossroad?”
I was not able to understand the solution given at first, so I tried to solve the problem on my own. Once I did, I was able to see what the Futility Closet solution was getting at. Certainly diagrams were needed to make sense of it all, and that is what I provided. See the Four Travelers Problem.
Reading Axios on Christmas Eve day 2017, I was struck by what appeared at first to be a strange graph showing preferences for Christmas movies divided between men and women. The thing that struck me as strange was the computation for the total votes: the percentages were the average of the men and women percentages. This, of course, is not how you average percentages. What was going on? See Strange Statistics.
Recently I viewed a startling video by Matt Parker about the Tupper Self-Referential Formula. It is a formula that visually represents itself when graphed at a specific location in the (x, y) plane. I found it difficult to fathom, so I looked it up on Wikipedia and Google. After reading different explanations, I finally think I have the idea. So thought I would add my version to the mix. See Tupper Self-Referential Formula.
I came across the following entry in the Futility Closet website that cried out for justification. “An arrangement of three mutually perpendicular planes, like those in the corner of a cube, have a pleasing property: They’ll reflect a ray of light back in the direction that it came from.” So the question is, why is this reflection property true? See Corner Reflectors.
All too frequently I come across the usual statements questioning why non-technical folks should bother studying math. A typical example is the Pythagorean Theorem. People say, “What good is that? I’ll never use it. So why bother?” Ah, the famous “utility” argument – as if everything worthwhile must be “useful.” I thought I would take this “useless” math example par excellence and show that, in fact, it harbors many of the best aspects of mathematics that anyone should find appealing. See the Pythagorean Theorem
Paul Krugman in his 26 July 2013 New York Times column “The Conscience of a Liberal” provided an analysis about why we are always being caught in the slow lane of a congested highway. I tried to fill in the gaps in his laconic explanation. See the Slow Lane Problem.
This is another puzzle from the Futility Closet that was originally from Henry Dudeney’s Canterbury Puzzles.
“A country baker sent off his boy with a message to the butcher in the next village, and at the same time the butcher sent his boy to the baker. One ran faster than the other, and they were seen to pass at a spot 720 yards from the baker’s shop. Each stopped ten minutes at his destination and then started on the return journey, when it was found that they passed each other at a spot 400 yards from the butcher’s. How far apart are the two tradesmen’s shops? Of course each boy went at a uniform pace throughout.”
See the Two Errand Boys.
Years ago during one of my many excursions into the history of mathematics I wondered how Mercator used logarithms in his map projection (introduced in a 1569 map) when logarithms were not discovered by John Napier (1550-1617) and published in his book Mirifici Logarithmorum Canonis Descriptio until 1614, three years before his death in 1617. The mystery was solved when I read a 1958 book by D. W. Waters which said Edward Wright (1561-1615) in his 1599 book Certaine Errors in Navigation produced his “most important correction, his chart projection, now known as Mercator’s.” Wright did not use logarithms explicitly but rather implicitly through the summing of discrete secants of the latitude as scale factors. But what really caught my attention in the Waters book was this arresting footnote: “Wright explained his projection in terms of a bladder blown up inside a cylinder, a very good analogy.” This article recounts my exploration of this idea. See Mercator Projection Balloon.
(Update 4/2/2022) Balloon Idea as Rubberband
Imagine my surprise when I realized Burkard Polster’s latest Mathologer post “The magic log wheel: How was this missed for 400 years?” involving a circular sliderule presented the logarithm effect as stretching a rubberband around a circle. This is essentially the balloon effect only sort of in reverse.