Setting aside my chagrin that the following problem was given to pre-university students, I initially found the problem to be among the daunting ones that offer little information for a solution. It also was a bit “inelegant” to my way of thinking, since it involved considering some separate cases. Still, the end result turned out to be unique and satisfying (Talwalkar’s Note 2 was essential for a unique solution, since the problem as stated was ambiguous).
“Kshitij from India sent me this problem from the 1994 India Regional Mathematics Olympiad.
‘A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is 15000. What are the page numbers on the torn leaf?’
Note 1: a ‘leaf’ means a single sheet of paper.
Note 2: the quoted problem is actual wording from the competition. But let me add an important detail: the book is numbered in the usual sequential way starting with the first page as page 1.” See the Missing Pages Puzzle.
This is a riff on a classic problem, given in Challenging Problems in Algebra.
“N. Bank and S. Bank are, respectively, the north and south banks of a river with a uniform width of one mile. Town A is 3 miles north of N. Bank, town B is 5 miles south of S. Bank and 15 miles east of A. If crossing at the river banks is only at right angles to the banks, find the length of the shortest path from A to B.
Challenge. If the rate of land travel is uniformly 8 mph, and the rowing rate on the river is 1 2/3 mph (in still water) with a west to east current of 1 1/3 mph, find the shortest time it takes to go from A to B. [The path across the river must still be perpendicular to the banks.]” See the River Crossing.
Here is another imaginative geometry problem from Catriona Shearer’s twitter account.
“What fraction of the largest square is shaded?”
See the Cascading Squares Problem.
The issue 7 of the Chalkdust mathematics magazine had an interesting geometric problem presented by Matthew Scroggs.
“In the diagram, ABDC is a square. Angles ACE and BDE are both 75°. Is triangle ABE equilateral? Why/why not?”
I had a solution, but alas, the Scroggs’s solution was far more elegant. See the Chalkdust Triangle Problem.
It is a bit presumptuous to think I could reduce the universe of mathematics to some succinct essence, but ever since I first saw a column in Martin Gardner’s Scientific American Mathematical Games in 1967, I thought his example illustrated the essential feature of mathematics, or at least one of its principal attributes. And he posed it in a way that would be accessible to anyone. I especially wanted to credit Martin Gardner, since the idea resurfaced recently, uncredited, in some attractive videos by Katie Steckles and James Grime. (This reminds me of the Borges idea that “eighty years of oblivion are perhaps equal to novelty”.) See the Essence of Mathematics.
Being born on February 29 I have always had an interest in the calendar and the mechanics of Leap Year. Since I am sure everyone knows about Leap Year, I will just rattle off a few trivia questions to stimulate the memory. Why was I excited about my birthday in 2000 when everyone knew it was a Leap Year, being 4 years after 1996? When I lived in Brazil, everyone referred to Leap Year as bissextile. What was that all about? After the Gregorian reform in 1582, how come George Washington’s mother recorded his birth in their family bible as 11 February 1731 when we say it is 22 February 1732 (whereas Abraham Lincoln’s mother recorded 12 February 1809 for her son, which we agree with)? See February 29.
Update (2/29/2020) Continue reading
I have almost completed my original goal of publishing articles I have written to myself over the last several years regarding matters mathematical (together with a sprinkling of more recent items). From the visit counts I can tell someone is reading them, but other than spam from porn and gambling sites and intrusions from Russian bots, I have received no feedback on the material in comments, nor via the more private venue of email: email@example.com.
Certain mysteries have arisen, such as the enormous hits on the Pool Party posting, whereas the More Pool post in a similar vein has received much less attention. I can guess the interest in the Three Jugs Problem and Three Jugs Problem Redux may have stemmed from the Bing and DuckDuckGo search results, but the Pool Party remains a mystery.
Even though I get a kick out of producing these articles for my own satisfaction, I wouldn’t mind hearing more views on my solutions and commentary, since I am quite rusty on these matters and welcome questions, corrections, and clarifications. I am also curious about further topics and postings I might consider. It seems that the Puzzles and Problems receive the most visits, whereas my own Curiosities and Questions the least, and the Math Inquiries somewhere in between. I have a few more ideas I might explore, but would welcome any suggestions that are within my ability to address.
This essay began as an effort to prove Tanya Khovanova’s statement in her article “The Annoyance of Hyperbolic Surfaces” that her crocheted hyperbolic surface had constant (negative) curvature. I discussed Khovanova’s article in my previous essay “Exponential Yarn”. What I thought would be a fairly straight-forward exercise turned into a more concerted effort as I concluded that her crocheted surface did not have constant curvature. However, I found additional references that supported her statement, so I was becoming quite confused. I looked at other, similar surfaces to try to understand the whole curvature situation. This involved a lot of tedious computations (with my usual plethora of mistakes) that proved most challenging. But then I realized where I had gone astray. To cover my ignorance I claimed my error stemmed from a subtle misunderstanding. Herewith is a presentation of what I found. See Bugles, Trumpets, and Beltrami.
(Update 4/6/2019) Continue reading
Tanya Khovanova’s recent blog post “The Annoyance of Hyperbolic Surfaces” about crocheting a hyperbolic surface added to the numerous examples of such activity, usually from knitting. Somehow this post caught my attention, in particular about the exponential growth of each added row and the fact that the resulting “surface” had constant negative curvature. I explored the exponential growth in this article and saved the mathematical exploration of the constant negative curvature for a later essay. See Exponential Yarn.
Over the years one of the subjects I return to periodically to study is Einstein’s Theory of Relativity, both the Special and General theories. Interest in the Special Theory focused on the derivation of the Lorentz transformations (or contractions). Why did objects appear with different lengths and clocks run at different speeds for observers moving relative to one another? Early on (late 60s) I came across a great explanation in the 1923 book by C. P. Steinmetz. He derived it from two general assumptions of special relativity: (1) that all motion is relative, the motion of the railway train relative to the track being the same as the motion of the track relative to the train, and (2) that the laws of nature, and thus the velocity of light, are the same everywhere. I did not follow his derivation completely, so I produced my own, which I will give here. See the Lorentz Transformation.