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.
I really was trying to stop including Catriona Shearer’s problems, since they are probably all well-known and popular by now. But this is another virtually one-step-solution problem that again seems impossible at first. Many of her problems entail more steps, but I am especially intrigued by the one-step problems.
“What’s the sum of the two marked angles?”
See Kissing Angles.
Since everyone by now who has any interest has gone directly to Catriona Shearer’s Twitter account for geometric puzzles, I was not going to include any more. But this one with its one-step solution is too fine to ignore and belongs with the “5 Problem” as one of the most elegant.
“Two squares sit on the hypotenuse of a right-angled triangle. What’s the angle?”
See the Two Block Incline Puzzle
(Update 4/26/2019) Continue reading
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?”
See the Parallelogram Problem
Here is another imaginative geometry problem from Catriona Shearer’s twitter account.
“What fraction of the largest square is shaded?”
See the Cascading Squares Problem.
This is another interesting problem from Catriona Shearer. She shows the following figure with a regular hexagon and rectangle.
“The area of the regular hexagon is 30. What’s the area of the rectangle?”
See the Hexagon-Rectangle Problem.
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.