This is an old puzzle from Catriona Agg that I found on BL’s Math Games website.
“All four triangles are equilateral. What fraction of the largest triangle is shaded?”
This problem turned out to be both easy and unexpectedly challenging, at least for me.
See Four Equilateral Triangles for solutions.
This is a puzzle from the A+Click site.
“There is a fault with the cruise control on Hank’s car such that the speed continuously and linearly increases with time. When he starts off the speed is set to exactly 60 mph. He is driving on a long straight route with the radio on at full blast and he is not paying any attention to his speed. After 3 hours he notices that his speed has now reached 80 mph. For how many miles did he drive above the state speed limit of 70 mph?
Answer Choices: 125 miles 112.5 miles 105 miles 99.5 miles”
See Unlawful Distance for solutions.
This is another nice problem from the 2025 Math Calendar.
“The two squares in the diagram have areas 9 and 16 and form two of the sides of a triangle with area 3. What is the area x of the blue triangle?”
Again, the result must be a number of a day in a month.
See the Two Squares Puzzle for a solution.
This is another Catriona Agg puzzle that I also found somewhat challenging.
“Four squares. What’s the angle?”
Visio showed me the answer fairly soon, but it took a bit to figure out a proof.
See the Rotating Square Problem for a solution.
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.
