Here is a simple *Futility Closet* problem from 2014.

“This unit square is divided into four regions by a diagonal and a line that connects a vertex to the midpoint of an opposite side. What are the areas of the four regions?”

See the Square Deal

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Here is a simple *Futility Closet* problem from 2014.

“This unit square is divided into four regions by a diagonal and a line that connects a vertex to the midpoint of an opposite side. What are the areas of the four regions?”

See the Square Deal

One of my favorite bloggers, Kevin Drum, decided to relieve the tedium of our current political anarchy by whacking the hornets’ nest of the high school mathematics curriculum, in particular the subject of plane geometry. You can tell from the tag list on my blog that I hold plane geometry in high regard and can’t let this gibe pass without some rebuttal, futile as it may be. Actually, I am not going to weigh in on the general issue of the current math curriculum that much, but rather make a few observations from my own experience over the years as it relates to Kevin’s post.

Here is a problem from *Five Hundred Mathematical Challenges* that I indeed found quite challenging.

**“Problem 235. **Two fixed points *A* and *B* and a moving point *M *are taken on the circumference of a circle. On the extension of the line segment *AM *a point *N *is taken, outside the circle, so that lengths MN = MB*. *Find the locus of *N.*”

Since one of the first hurdles I faced with this problem was trying to figure out what type of shape was being generated, I thought I would omit my usual drawings illustrating the problem statement. There turned out to be a lot of cases to consider, but the result was most satisfying. I also included the case when *N* is inside the circle. Again Visio was my main tool to handle all the examples with the concomitant requirement to prove whatever Visio suggested.

See the Curve Making Puzzle

This is another delightful Brainteaser from the *Quantum* math magazine.

“All the vertices of a polygonal line ABCDE lie on a circumference (see the figure), and the angles at the vertices B, C, and D are each 45°.

Prove that the area of the blue part of the circle is equal to the area of the yellow part. (V. Proizvolov)”

I especially liked this problem since I was able to find a solution different from the one given by *Quantum*. Who knows how many other variations there might be.

See the Circle-Halving Zigzag Problem

In my search for problems I decided to purchase Dan Griller’s GCSE problem book mentioned in the Cube Roots Problem. I am still a bit confused about the purpose of the GCSE exam and who it is for, since the other problems in Griller’s book are often as challenging or more so than the cube roots problem. It is hard to believe students not pursuing college level degrees could solve these problems. (Grades 8 and 9 referred to in the subtitle of the book must indicate something other than US grades 8 and 9, since the exams are aimed at 16 year-olds, not 13 and 14 year-olds.)

Supposedly the problems in Griller’s book are nominally arranged in increasing order of difficulty from problem 1 to problem 75. However it seemed to me that there were challenging problems scattered throughout and the last problem was not all that much harder than earlier ones. And many of them had a whiff of Coffin Problems—they seemed impossible at first (Problem 44: Construct a 67.5° angle!). I don’t know how many problems are on the exam or how long the exam is, but anyone taking a timed exam does not have the leisure to mull over a problem. The student only has a few minutes to come up with an approach and clever insights are rare under the circumstances. Anyway, here is the last problem in the book.

**“Problem 75.** A square pond of side length 2 metres is to be surrounded by twelve square paving stones of side length 1 metre.

(a) The first design is constructed with a circle whose centre coincides with the centre of the pond. Calculate exactly the total dark grey area for this design.

(b) The second design is similar. Calculate exactly the total dark grey area for this second design.”

See the Pool Paving Problem

This is a problem from the UKMT Senior Challenge for 2019. (It has been slightly edited to reflect the colors I added to the diagram.)

“The edge-length of the solid cube shown is 2. A single plane cut goes through the points Y, T, V and W which are midpoints of the edges of the cube, as shown.

What is the area of the cross-section?

A_√3_____B_3√3_____C_6_____D_6√3_____E_8”

This is a nice Brainteaser from the *Quantum* math magazine.

“Line segment MN is the projection of a circle inscribed in a right triangle ABC onto its hypotenuse AB. Prove that angle MCN is 45°.”

See the Circle Projection Problem.

I found an interesting geometric statement in a paper of Glen Van Brummelen cited in the online MAA January 2020 issue of *Convergence*:

“For instance, Abū’l-Wafā’ describes how to embed an equilateral triangle in a square, as follows: extend the base GD by an equal distance to E. Draw a quarter circle with centre G and radius GB; draw a half circle with centre D and radius DE. The two arcs cross at Z. Then draw an arc with centre E and radius EZ downward, to H. If you draw AT = GH and connect B, H, and T, you will have formed the equilateral triangle.”

So the challenge is to prove this statement regarding yet another fascinating appearance of an equilateral triangle.

See the Triangle of Abū’l-Wafā’

This turned out to be a challenging geometric problem from Poo-Sung Park posted at the Twitter site #GeometryProblem

**“Geometry Problem 92: **What is the ratio of a:b?”

See the Envelope Puzzle

Here is another problem from the *Quantum* magazine, only this time from the “Challenges” section (these are expected to be a bit more difficult than the Brainteasers).

“A quadrangle is inscribed in a parallelogram whose area is twice that of the quadrangle. Prove that at least one of the quadrangle’s diagonals is parallel to one of the parallelogram’s sides. (E. Sallinen)”