# 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

# 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.

(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?”

See the Impossible Car Riddle

# Lucky 7 Problem

This interesting problem comes from Colin Hughes at the Maths Challenge website.

Problem
Prove that for any number that is not a multiple of seven, then its cube will be one more or one less than a multiple of 7.”

# 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?”

See the Tandem Bicycle Puzzle.

# Cube Slice Angle Problem

This is from the UKMT Senior Challenge of 2004.

“L, M, and N are midpoints of a skeleton cube, as shown. What is the value of angle LMN?

_____A_90°_____B_105°_____C_120°_____D_135°_____E_150°”

# 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.

# Marching Cadets and Dog Problem

In my search for new problems I came across this one from Martin Gardner:

“A square formation of Army cadets, 50 feet on the side, is marching forward at a constant pace [see Figure]. The company mascot, a small terrier, starts at the center of the rear rank [position A in the illustration], trots forward in a straight line to the center of the front rank [position B], then trots back again in a straight line to the center of the rear. At the instant he returns to position A, the cadets have advanced exactly 50 feet. Assuming that the dog trots at a constant speed and loses no time in turning, how many feet does he travel?”

Gardner gives a follow-up problem that is virtually impossible:

“If you solve this problem, which calls for no more than a knowledge of elementary algebra, you may wish to tackle a much more difficult version proposed by the famous puzzlist Sam Loyd. Instead of moving forward and back through the marching cadets, the mascot trots with constant speed around the outside of the square, keeping as close as possible to the square at all times. (For the problem we assume that he trots along the perimeter of the square.) As before, the formation has marched 50 feet by the time the dog returns to point A. How long is the dog’s path?”

See the Marching Cadets and Dog Problem.

# Logging Problem

This is a delightful little problem from Dick Hess that exercises one’s basic facility with logarithms:

“Define x as

Find an expression for

in terms of x where the only constants appearing are integers.”

See the Logging Problem