Here is an entertaining puzzle from Futility Closet.
“By Wikimedia user Efbrazil. Begin at the star. The number at your current position tells you the number of blocks that your next jump must span. All jumps must be orthogonal. So, for example, your first jump must take you to the 1 in the lower left corner or the 2 in the upper right. What sequence of jumps will return you to the star?”
See A Number Maze for solutions
This is a somewhat unusual problem from Presh Talwalkar. It involves proving a student’s homework problem is impossible.
“I came across a homework problem described as “scary” on Reddit AskMath. You need to fill in the number sentences using the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once.
You should try a few possibilities to see why this is a challenging question. And do not waste too much time because the exercise is literally impossible! The challenge is, can you prove no solution exists?”
See Impossible Homework for a solution.
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 a wicked variation of the ant problem on a stick by Peter Winkler.
“Twenty-four ants are randomly placed on a circular track of length 1 meter; each ant faces randomly clockwise or counterclockwise. At a signal, the ants begin marching at 1 cm/sec; when two ants collide they both reverse directions. What is the probability that after 100 seconds, every ant finds itself exactly where it began?”
See Circular Ant Problem for solutions.
I think this turned out to be an even trickier problem than Alex Bellos thought.
“Tricky triangle This one was sent in by a reader, aged 85, who first saw it in 1960. He is a roboticist who passed through Harvard, Princeton, Stanford and IBM. He says it is his favourite puzzle. ‘I’ve given this puzzle to perhaps 100 people. Over 80% have no idea how to solve it.’ What is the length of AD, the dashed line?”
See Tricky Triangle for solutions.
This is a classic puzzle from Boris Kordemsky’s 1972 Moscow Puzzles.
“Purrer has decided to take a nap. He dreams he is encircled by 13 mice: 12 gray and 1 white. He hears his owner saying: “Purrer, you are to eat each thirteenth mouse, keeping the same direction. The last mouse you eat must be the white one.” Which mouse should he start from [eat first]?”
See Cat and Mice for a solution.
The Josephus Problem
This famous Josephus Problem presented on Youtube is somewhat different from the Cat and Mice puzzle, but still has similarities. An article by Jay Bennett discussing the problem was published in Popular Mechanics in 2016.
Penn and Teller – Spelling Cards
It turns out that Penn and Teller have performed another magic trick recently that is based on mathematical principles and so is more or less self-working. It is a more complicated version of the Cat and Mice puzzle, which I have dubbed the “Spelling Cards” trick. Continue reading
This is a fun puzzle from John Bassey at Puzzle Sphere.
“The diagram shows a heptagon with three circles on each side. Some circles already have the numbers 8 to 14 filled in, while the remaining circles need to be filled with the numbers 1 to 7. Each circle must contain one number, and the sum of the numbers in every set of three circles along a line must be the same. Arrange the numbers!!!”
See Fill in the Blanks 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
Thanks to Futility Closet I discovered a new source of math puzzles: A+Click.
“A+ Click helps students become problem solvers. Free, without ads, no calculators, and without signing-up. The website features a graduated set of 16,000+ challenging problems for students in grades one through twelve, starting from the very simple to the extremely difficult. … The questions concentrate on understanding, spatial reasoning, usefulness, and problem solving rather than math rules and theorems. The problems include a short description and an illustration to help problem solvers visualize the model. The problems can be solved within one minute and without using a calculator.”
My only quibble with “The questions concentrate on understanding, spatial reasoning, usefulness, and problem solving rather than math rules and theorems.” is that by keeping explicit math notation and concepts to a minimum, the use of symbolic algebra and calculus is muted and there is a whiff of the medieval reliance on mental verbal agility rather than the power of the new mathematics.
Still the problems are imaginative and challenging. Here is a good example.
“The rear tires of my car wear out after 40,000 miles, while the front tires are done after 20,000 miles. Estimate how many miles I should drive before the tires (front and rear) are rotated to drive the maximal distance.
Answer Choices: 15,000 miles 12,000 miles 13,333 miles 16,667 miles”
(I admit solving these under a minute is a challenge, at which I often failed. Ignoring time constraints allows for greater care and a more thorough mulling over the intricacies of the problem. Yes, those who have mastered math can solve problems faster than those who have not, but real mastery of math requires an inordinate attention to details, and that requires time.)
See Tire Wear for solutions.
