Monthly Archives: May 2025

Cheshire Cat Paradigm

I have been meaning to focus on this aspect of mathematics for some time.  It is a topic I elaborated in my “Angular Momentum” post. But I also think it has something to do with the difficulties that normal folks have with elementary math, in particular, numbers.  I thought I would dub it the Cheshire Cat Paradigm, involving the Cheshire Cat’s grin.

See the Cheshire Cat Paradigm.

(Update 6/7/2025) Contra Concrete Algebra

A recent posting has prompted me to address yet again my concerns about tying the learning of algebra so tightly to concrete objects and manipulations.

See Contra Concrete Algebra.

Wittenbauer’s Parallelogram

This is a lovely result from Futility Closet.

“Draw an arbitrary quadrilateral and divide each of its sides into three equal parts. Draw a line through adjacent points of trisection on either side of each vertex and you’ll have a parallelogram.

Discovered by Austrian engineer Ferdinand Wittenbauer.”

Find a proof.

See Wittenbauer’s Parallelogram for a solution.

Two Squares in a Circle

This puzzle, from another set of seven challenges assembled by Presh Talwalkar, turned out to be very challenging for me.

“This is a fun problem I saw on Reddit AskMath. A circle contains two squares with sides of 4 and 2 cm that overlap at one point, as shown. What is the area of the circle?”

This took me quite a while to figure out, but I relied on another problem I had posted earlier.

Answer.

See Two Squares in a Circle for solutions.

Whose Bullet?

Here is a probability problem from BL’s Weekly Math Games.  Normally I am not a fan of such problems, but this one seemed fairly straight-forward for a change.

“I hit the target 75% of the time. You hit the target 25% of the time.  We aim at the same time, and only one bullet hits.  What’s the probability it came from me?

Now as for this puzzle, it would be tempting to think that I am 3 times as good at hitting the target, but I am not!”

Answer.

See Whose Bullet for a solution