This is a challenging problem from Mathematical Quickies (1967).
“Evaluate the infinite product:”
I came up with a motivated solution using some standard techniques from calculus. Mathematical Quickies had a solution that did not employ calculus, but one which I felt used unmotivated tricks. See the Infinite Product Problem.
I came across this problem in Alfred Posamentier’s book, but I remember I had seen it a couple of places before and had never thought to solve it. At first, it seems like magic.
In any convex quadrilateral (non-intersecting sides) inscribe a second convex quadrilateral with its vertices on the midpoints of the sides of the first quadrilateral. Show that the inscribed quadrilateral must be a parallelogram.
See the Magic Parallelogram.
Here is a problem from the famous (infamous?) Putnam exam, presented by Presh Talwalkar. Needless to say, I did not solve it in 30 minutes—but at least I solved it (after making a blizzard of arithmetic and trigonometric errors).
“Today’s problem is from the 1978 test, problem B1 (the easiest of the second set of problems). A convex octagon inscribed in a circle has four consecutive sides of length 3 and four consecutive sides of length 2. Find the area of the octagon.”
My solution is horribly pedestrian and fraught with numerous chances for arithmetic mistakes to derail it, which happened in spades. As I suspected, there was an elegant, “easy” solution (as demonstrated by Talwalkar)—once you thought of it! Again, this is like a Coffin Problem. See the Putnam Octagon Problem.
This is one of H. E. Dudeney’s train puzzles.
“Two railway trains, one four hundred feet long and the other two hundred feet long, ran on parallel rails. It was found that when they went in opposite directions they passed each other in five seconds, but when they ran in the same direction the faster train would pass the other in fifteen seconds. A curious passenger worked out from these facts the rate per hour at which each train ran. Can the reader discover the correct answer? Of course, each train ran with a uniform velocity.”
See Two Trains – Passing in the Night.
It was reading Peter Hoffmann’s 2012 book Life’s Ratchet that drove home the role of determinism in biological processes, which he characterizes as a ratchet, a process that filters random behavior into a particular “purposeful” direction. Since Hoffmann is a biophysicist, his presentation is heavily guided by the physical principles of energy conversion, thermodynamics, and entropy, which makes for a fresh approach to a traditionally biological subject. The startling thing Hoffmann’s book introduced me to was the subject of molecular machines or motor proteins. These were amazing engines that harnessed the chemical and physical energy within a cell to act like miniature workers hauling materials around and constructing other molecules. The intelligent design crowd would go bonkers. See Chemical Determinism – Motor Proteins
James Tanton had another interesting puzzle on Twitter.
“Points P and Q each move counterclockwise on a circle, uniform speed, one revolution per minute. At each instant, segment PQ is translated so that P is at the origin. Let Q’ be the image of Q. What curve is traced by the points Q’?”
See the Tandem Circles.
This is one the best articles I have read on gerrymandering regarding its political import, and of course it is by one of the most articulate mathematicians, Jordan Ellenberg:
“Fixing partisan gerrymandering requires some technical calculations. That’s why we filed a mathematicians’ brief to better define the problem—and the solution.”
See Gerrymandering at SCOTUS. (You will have to read the article to understand the picture.)
(Update 4/8/2019) Continue reading
One of the all-time examples of chance intervening in history is Christopher Columbus’s putative discovery of America. Moreover, the legend of this discovery is filled with erroneous information that was traditionally foisted upon unsuspecting elementary school children. One of the most egregious errors was the assertion that Columbus was trying to prove the earth was round and not flat. I had a picture book when I was young that showed sailors tumbling off the edge of a flat earth.
I first came upon the demythologizing of the Columbus legend from reading Isaac Azimov’s anthologized 1962 column “The Shape of Things”. His tale is so well-written, that I want to include it in its entirety. I have augmented it with some more detailed footnotes and illustrations.
See Columbus and the Irony of Chance.
This essay introduces a topic I have been thinking about for a number of years. It also may allow me to connect the math impulse to a wider range of thoughts than just those based on math or even science.
It all begins with the perennial question of “why” that drives our curiosity about the nature of things and how various situations came about, such as our physical universe, our biology, the origin of life, or historical events. The explanations are usually couched in terms of causal links: such and such happened because some other thing happened. In the physical sciences we think the causal links follow certain physical, chemical, or biological laws that we provisionally hypothesize. In the historical realm we think there are still causes, such as the physical environment (geography, climate, weather, etc.) or the imprint of individuals. But the historical chains of events are often disrupted by chance and coincidences, and some supposed links degenerate into imagined connections or associations.
In the future I plan to write a number of essays that explore and illustrate these ideas. See Causality, Chance, and Connections.
I was astonished that this problem was suitable for 8th graders. First of all the formula for the volume of a cone is one of the least-remembered of formulas, and I certainly never remember it. So my only viable approach was calculus, which is probably not a suitable solution for an 8th grader.
Presh Talwalkar: “This was sent to me as a competition problem for 8th graders, so it would be a challenge problem for students aged 12 to 13. When a conical bottle rests on its flat base, the water in the bottle is 8 cm from its vertex. When the same conical bottle is turned upside down, the water level is 2 cm from its base. What is the height of the bottle? (Note “conical” refers to a right circular cone as is common usage.) I at first thought this problem was impossible. But it actually can be solved. Give it a try and then watch the video for a solution.”
See the Conical Bottle Problem.