Author Archives: Jim Stevenson

Trains – Pickleminster to Quickville

This is another train puzzle by H. E. Dudeney. This one has some hairy arithmetic.

“Two trains, A and B, leave Pickleminster for Quickville at the same time as two trains, C and D, leave Quickville for Pickleminster. A passes C 120 miles from Pickleminster and D 140 miles from Pickleminster. B passes C 126 miles from Quickville and D half way between Pickleminster and Quickville. Now, what is the distance from Pickleminster to Quickville? Every train runs uniformly at an ordinary rate.”

Answer.

See Trains – Pickleminster to Quickville for a solution.

Mountain Houses Problem

It is always fascinating to look at problems from the past. This one, given by Thomas Whiting himself, is over 200 years old from Whiting’s 1798 Mathematical, Geometrical, and Philosophical Delights:

“Question 2, by T. W. from Davison’s Repository.
There are two houses, one at the top of a lofty mountain, and the other at the bottom; they are both in the latitude of 45°, and the inhabitants of the summit of the mountain, are carried by the earth’s diurnal rotation, one mile an hour more than those at the foot.

Required the height of the mountain, supposing the earth a sphere, whose radius is 3982 miles.”

Answer.

See the Mountain Houses Problem for solutions.

Consecutive Product Square

This problem from Colin Hughes at Maths Challenge is a most surprising result that takes a bit of tinkering to solve.

“Problem
We can see that 3 x 4 x 5 x 6 = 360 = 19² – 1. Prove that the product of four consecutive integers is always one less than a perfect square.”

The result is so mysterious at first that you begin to understand why the ancient Pythagoreans had a mystical relationship with mathematics.

See the Consecutive Product Square.

(Update 11/12/2020) Generalization and Visual Proof
Continue reading →

Mathematics, And The Excellence Of The Life It Brings

I am a regular reader of Ash Jogalekar’s blog Curious Wavefunction, but I found my way to his latest via the eclectic website 3 Quarks Daily, also highly recommended. I could not resist the title, “Mathematics, And The Excellence Of The Life It Brings”. The entirety of the post was about the mathematician Shing-Tung Yau’s recent memoir, The Shape of a Life, but Jogalekar’s introductory remarks about his personal involvement with mathematics stirred so many personal recollections of my own, that I thought I would provide an excerpt, followed by my own comments. Furthermore, he also addresses in passing the perennial question of whether math is invented or discovered.

See Math and the Excellence of Life

(Update 8/9/2021) Jogalekar’s story about his embracing math and the effect Simmon’s topology book had on him is even more amazing than I thought.  Throughout his younger years he had always been labeled “bad at math” and did poorly in school.  But a teacher and Simmon’s book changed all that.  He explains in a recent article in 3QuarksDaily, which I also provide here. 

I can’t help singling out a section where he, too, extols the significance and importance of high school geometry (see my post “Down with Geometry”):

“… Purely through accident at this time, I had gotten my hands on a book on topology, a subject that I had become mildly interested in because of its deep connections to geometry; interestingly, while I was rather abysmal at algebra in school, I always did well with geometry because I was good at visualization.  …

The topology book and the professor completely changed my outlook and saved me. I started doing well and tackling advanced topics and started to love math. I also got interested in physics and did well. Most importantly, I started appreciating the beauty of math. Over time I found that people interested in mathematics are generally of two kinds, although there’s some overlap: there are those who really enjoy mathematical puzzles and puzzle-like problems, relishing the raw process of problem-solving. Then there are others who simply enjoy the abstract nature of proofs and the connections between different topics: I am definitely part of this second group. In fact, another revelation I had was that most of the high school curriculum needed the students to be good at the former skill and had no appreciation of the latter, thus simply weeding out students like myself who wanted to understand the big picture and see the connections rather than just become adept at problem-solving.”

I confess I share this view and find it somewhat ironic that my website has devolved into a problem-solving source.  I have tried to show the wider picture of fascinating connections, but that often takes more skill and time than I currently possess.

Maximum Product

This 2007 four-star problem from Colin Hughes at Maths Challenge is definitely a bit challenging.

“Problem
For any positive integer, k, let Sk = {x1, x2, … , xn} be the set of [non-negative] real numbers for which x1 + x2 + … + xn = k and P = x1 x2 … xn is maximised. For example, when k = 10, the set {2, 3, 5} would give P = 30 and the set {2.2, 2.4, 2.5, 2.9} would give P = 38.25. In fact, S10 = {2.5, 2.5, 2.5, 2.5}, for which P = 39.0625.

Prove that P is maximised when all the elements of S are equal in value and rational.”

I took a different approach from Maths Challenge, but for me, it did not rely on remembering a somewhat obscure formula. (I don’t remember formulas well at my age—only procedures, processes, or proofs, which is ironic, since at a younger age it was just the opposite.) It is also clear from the Maths Challenge solution that the numbers were assumed to be non-negative.

See Maximum Product.

Impossible Car Riddle

This is another intriguing problem from Presh Talwalkar.

“A car travels 75 miles per hour (mph) downhill, 60 mph on flat roads, and 50 mph uphill. It takes 3 hours to go from town A to B, and it takes 3 hours and 30 minutes for return journey by the same route. What is the distance in miles between towns A and B?”

Answer.

See the Impossible Car Riddle for solutions.

Tandem Bicycle Puzzle

A glutton for punishment I considered another Sam Loyd puzzle:

“Three men had a tandem and wished to go just forty miles. It could complete the journey with two passengers in one hour, but could not carry the three persons at one time. Well, one who was a good pedestrian, could walk at the rate of a mile in ten minutes; another could walk in fifteen minutes, and the other in twenty. What would be the best possible time in which all three could get to the end of their journey?”

Answer.

See the Tandem Bicycle Puzzle for a solution.

Magic Pythagorean Circle

This statement showed up recently at Futility Closet and I found it to be another one of those magical results that seemed so surprising. I don’t recall ever seeing this before.

“The radius of a circle inscribed in a 3-4-5 triangle is 1.
(In fact, the inradius of any Pythagorean triangle is an integer.)”

(A Pythagorean triangle is a right triangle whose sides form a Pythagorean triple.) Futility Closet left these remarkable statements unproven, so naturally I felt I had to provide a proof.

See Magic Pythagorean Circle

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