Two Trains – Passing in the Night

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.”

Answer.

See Two Trains – Passing in the Night for a solution.

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Chemical Determinism – Motor Proteins

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

(Update 8/24/2025)  Origin of Life — Probably Not by Chance  Continue reading →

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Tandem Circles

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’?”

Answer.

See the Tandem Circles for an answer.

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Gerrymandering at SCOTUS

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.)

(Updates 4/8/2019, 6/27/2019, 8/27/2022)  Continue reading →

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Columbus and the Irony of Chance

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.

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Causality, Chance, and Connections

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.

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Conical Bottle Problem

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.”

Answer.

See the Conical Bottle Problem for solutions.

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Regiomontanus 1471 Problem

This is an old problem I had seen before. Here is David Wells’s rendition:

“Johannes Müller, named Regiomontanus after the Latin translation of Körnigsberg, his city of birth, later made famous by Euler, proposed this problem in 1471. … it is usually put in this form …: From what distance will a statue on a plinth appear largest to the eye [of a mouse!]? If we approach too close, the statue appears foreshortened, but from a distance it is simply small.”

I have added height numbers in feet for concreteness (as well as the mouse qualification, since the angles are measured from ground level). So the problem is to find the distance x such that the angle is maximal.

Answer.

See the Regiomontanus 1471 Problem for solutions.

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Parallelogram Problem

Catriona Shearer has come up with another challenging but elegant geometric problem. In some ways, it is similar to the famous Russian Coffin Problems that have an obvious solution—once you see it—but initially seem impenetrable. I really marvel at Catriona Shearer’s ability to come up with these problems.

“What’s the area of the parallelogram?”

Answer.

See the Parallelogram Problem for a solution.

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Who Gets To Be Called A Mathematician?

A mathematics friend of mine just sent me this link to a 2017 posting by “recovering mathematician” Junaid Mubeen that I found most apt. If you haven’t seen it yet, take a look.

“[M]athematics need not be situated at the extremes of established knowledge. We can all revel in problems whose solutions are known. Even when humankind has exhausted its capacity to extend its collective knowledge base, as individuals our ignorance is what keeps our mathematical instincts aflame. Problem solving lies between the boundaries of what we know and what we seek. This sweet spot is where we all—novices and experts alike—get to bend and twist what we know to forge new truths for ourselves. Who cares if our discoveries are already known to the rest of the world (or machines, for that matter)? The satisfaction of finding my own solution, of pushing through my own knowledge limits, is as enthralling as the pursuit of ‘new’ proofs promised by research mathematics. Let the machines come; mathematics does not belong to the omnipotent.”

This view echoes my feeling that we can enjoy performing music even if we can’t compose it. See Who Gets To Be Called A Mathematician.

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