Tag Archives: trigonometry

Geometric Puzzle Mayhem

I was really trying to avoid getting pulled into more addictive geometric challenges from Catriona Shearer (since they can consume your every waking moment), but a recent post by Ben Orlin, “The Tilted Twin (and other delights),” undermined my intent. As Orlin put it, “This is a countdown of her three favorite puzzles from October 2019” and they are vintage Shearer. You should check out Olin’s website since there are “Mild hints in the text; full spoilers in the comments.” He also has some interesting links to other people’s efforts. (Olin did leave out a crucial part of #1, however, which caused me to think the problem under-determined. Checking Catriona Shearer’s Twitter I found the correct statement, which I have used here.)

I have to admit, I personally found the difficulty of these puzzles a bit more challenging than before (unless I am getting rusty) and the difficulty in the order Olin listed. Again, the solutions (I found) are simple but mostly tricky to discover. I solved the problems before looking at Olin’s or others’ solutions.

See the Geometric Puzzle Mayhem.

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.

Six Squares Problem

This is a problem from the UKMT Senior Challenge for 2001. (It has been slightly edited to reflect the colors I added to the diagram.)

“The [arbitrary] blue triangle is drawn, and a square is drawn on each of its edges. The three green triangles are then formed by drawing their lines which join vertices of the squares and a square is now drawn on each of these three lines. The total area of the original three squares is A1, and the total area of the three new squares is A2. Given that A2 = k A1, then

_____A_ k = 1_____B_ k = 3/2_____C_ k = 2_____D_ k = 3_____E_ more information is needed.”

I solved this problem using a Polya principle to simplify the situation, but UKMT’s solution was direct (and more complicated).

Answer.

See the Six Squares Problem for solutions.

Triangle Acute-Angle Problem

Catriona Shearer retweeted the following problem from Antonio Rinaldi‏ @rinaldi6109

“My little contribution to @Cshearer41 October 7, 2018
A point D is randomly chosen inside the equilateral triangle ABC. Determine the probability that the triangle ABD is acute-angled.”

Answer.

See Triangle Acute-Angle Problem for a solution.

Parallelogram Cosine Problem

Another challenging problem from Presh Talwalkar. I certainly could not have solved it on a timed test at the age of 16.

“One Of The Hardest GCSE Test Questions – How To Solve The Cosine Problem

Construct a hexagon from two congruent parallelograms as shown. Given BP = BQ = 10, solve for the cosine of PBQ in terms of x.

This comes from the 2017 GCSE exam, and it confused many people. I received many requests to solve this problem, and I thank Tom, Ben, and James for suggesting it to me.”

Answer.

See the Parallelogram Cosine Problem for solutions.

Square Wheels

I came across the following problem from an Italian high school exam on the British Aperiodical website presented by Adam Atkinson:

“There have been various stories in the Italian press and discussion on a Physics teaching mailing list I’m accidentally on about a question in the maths exam for science high schools in Italy last week. The question asks students to confirm that a given formula is the shape of the surface needed for a comfortable ride on a bike with square wheels.

What do people think? Would this be a surprising question at A-level in the UK or in the final year of high school in the US or elsewhere?”

I had seen videos of riding a square-wheeled bicycle over a corrugated surface before, but I had never inquired about the nature of the surface. So I thought it would be a good time to see if I could prove the surface (cross-section) shown would do the job. See Square Wheels.

(Update 9/14/2023)  Square Bridge That Rolls!

This is an incredible application of the rolling square wheels idea described on Matt Parker’s Stand-up Maths Youtube website.  It also demonstrates the difference between engineering and pure math.  The engineers had to solve some challenging problems to adapt the theoretical math to a practical application.  And such solutions are always required under tight time constrictions.  Engineering certainly is a noble profession.

Complex Numbers – Geometric Viewpoint

This may be a futile attempt at an elementary introduction to complex variables by emphasizing their geometric properties. The elementary part is probably undermined by an initial discussion of field extensions and a necessary reference to trigonometry. Hopefully, the suppression of the explicit use of complex powers of Euler’s constant e until the very end will allow the geometric ideas to have center stage. A primary goal of the essay is to realize that complex polynomials involve sums of circles in the plane. The image of real polynomials as wavy curves in the plane is misleading for an understanding of complex behavior. See Complex Numbers – Geometric Viewpoint.

Geometric Puzzle Medley

This is a collection of simple but elegant puzzles, mostly from a British high school math teacher Catriona Shearer, for which I thought I would show solutions (solutions for a number of them had not been posted yet on Twitter at the time of writing).  See the Geometric Puzzle Medley.

Apparently Catriona Shearer creates these problems herself, which shows an especially gifted talent. Ben Olin, of Math with Bad Drawings fame, had an interesting interview with Ms. Shearer. The reason for the interest in her work becomes evident the more of her geometry problems one sees. They are especially elegant and minimalist, and often have simple solutions, as exemplified by the “5 Problem” or “Shear Beauty” problem illustrated here. Words, such as “beauty” and “elegance”, are often bandied about concerning various mathematical subjects, but as with any discussion of esthetics, the efforts at explanation usually fall flat. Shearer’s problems are one of the best examples of these ideas I have ever seen. If you contemplate her problems and even solve them, you will understand the meaning of these descriptions.

One of the key aspects of mathematics is often its “hidden-ness” (some would say “opacity” or “incomprehension”). Her problems appear to have insufficient information to solve. But as you look at the usually regular figures, you see that there are inherent rigid constraints that soon yield specific information that leads to a solution. This discovery is akin to the sensation of discovering Newton’s mathematical laws underlying physical reality. It is the essence of one of the joys of mathematics.

Continue reading →

Perspective Map

A number of recent puzzles have involved perspective views of objects. I had never really explored the idea of a perspective map in detail. So some of the properties associated with it always seemed a bit vague to me. I decided I would derive the mathematical equations for the perspective or projective map and see how its properties fell out from the equations. With this information in hand I then addressed some questions I had about the article “Dürer: Disguise, Distance, Disagreements, and Diagonals!” by Annalisa Crannell, Marc Frantz, and Fumiko Futamura concerning a controversy over Albrecht Dürer’s woodcut St. Jerome in His Study (1514). And finally, I read somewhere that a parabola under a perspective map becomes an ellipse, so I was able to show that as well. See the Perspective Map.

(Update 7/1/2019) Continue reading →

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