This surprising, but simple, puzzle is from the 12 April MathsMonday offering by MEI, an independent curriculum development body for mathematics education in the UK.
“In the diagram various regular polygons, P, have been drawn whose sides are tangents to a circle, C. Show that for any regular polygon drawn in this way:”
(Given that the polygons approximate the circle in the limit, it would not be surprising that this relationship would hold—in the limit. It is surprising that it should be true for every regular polygon that circumscribes the circle.)
See the Area vs. Perimeter Puzzle
These two interesting problems were posed on MEI’s MathsMonday site on 3 February 2020 and 2 March 2020, respectively. MEI and readers posted various approaches, but I used a method suggested by another problem whose origin I no longer recall.
See the Nested Polygons Puzzle
This simple-appearing problem is from the 17 August 2020 MathsMonday offering by MEI, an independent curriculum development body for mathematics education in the UK.
“The diagram shows an equilateral triangle in a rectangle. The two shapes share a corner and the other corners of the triangle lie on the edges of the rectangle. Prove that the area of the green triangle is equal to the sum of the areas of the blue and red triangles. What is the most elegant proof of this fact?”
Since the MEI twitter page seemed to be aimed at the high school level and the parting challenge seemed to indicate that there was one of those simple, revealing solutions to the problem, I spent several days trying to find one. I went down a number of rabbit holes and kept arriving at circular reasoning results that assumed what I wanted to prove. Visio revealed a number of fascinating relationships, but they all assumed the result and did not provide a proof. I finally found an approach that I thought was at least semi-elegant.
See the Diabolical Triangle Puzzle
(Update 1/30/2021) New MEI Solution