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.
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.
See Two Squares in a Circle for solutions.
For me this turned out to be sort of a challenging problem from the 2025 Math Calendar.
“Given equal line segments AB = CD, what is angle θ in degrees?”
See Elusive Angle for a solution
This is another intimidating puzzle from Presh Talwalkar:
“Thanks to Eric from Miami for suggesting this problem and sending a solution!
From a 5th grade Chinese textbook: In the quadrilateral ABCD, angle A = 90°, angle ABD = 40°, angle BDC = 5°, angle C = 45°, and the length of AB is 6. Find the area of the quadrilateral ABCD.”
See the Chinese Quadrilateral Puzzle for solutions.
This is an interesting problem from the Canadian Mathematical Society’s 2001 Olymon.
“Suppose that XTY is a straight line and that TU and TV are two rays emanating from T for which XTU = UTV = VTY = 60º. Suppose that P, Q and R are respective points on the rays TY, TU and TV for which PQ = PR. Prove that QPR = 60º.”
See the Ubiquitous 60 Degree Problem
This is a fairly straight-forward problem from the 1999 AIME problems.
“The two squares shown share the same center O and have sides of length 1. The length of AB is 43/99 and the area of octagon ABCDEFGH is m/n where m and n are relatively prime positive integers. Find m + n.”
See the Another Octagonal Area Problem for solutions.
Since Twitter (now X) is no longer public, I was afraid I would not have access to new Catriona Agg puzzles, but she has put them up on Instagram, which is partially available to the public. I managed to find a half dozen interesting new brain ticklers.
Coming across this classic geometric paradox recently in Futility Closet motivated me to write down its solution in detail.
“Where did the empty square come from?”
In any case, this is the canonical example for why I avoid visual geometric proofs—you can be so easily fooled. Real proofs require plane or analytic geometry arguments.
See the Classic Geometry Paradox
(Update 9/14/2024) Penn & Teller – Fool Us – Magic Trick
Continue reading
James Tanton asked to prove the following surprising property of a right triangle and its circumscribed and inscribed circles.
“Every triangle is circumscribed by some circle of diameter D, say, and circumscribes another circle of smaller diameter d. For a right triangle, d + D equals the sum of two side lengths of the triangle. Why?”
Here is another problem from the “Challenges” section of the Quantum magazine.
“Inside a circle there are two intersecting circles. One of them touches the big circle in point A, the other in point B. Prove that if segment AB meets the smaller circles at one of their common points, then the sum of their radii equals the radius of the big circle. Is the converse true? (A. Vesyolov)”
