This is an infinite series from the 2007 Graduate Record Exam (GRE) practice manual.
(A) e___(B) 2e___(C) (e + 1)(e – 1)___(D) e2___(E) ∞
See Challenging GRE Sum for a solution.
Challenging GRE Sum
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Conical Mountain Climb
This is a fun problem (#1455) from A+ Click.
“Gerry and Jane are at the foot of a conical-shaped mountain with a direct way to the peak of 8 km and the foot circle’s diameter of 8 km. They walk from point A to the diametrically opposite point B by the shortest route. Find the distance c from the foot of the mountain to the highest point of their route C.
Answer Choices:__2.34 km__2.66 km__3.34 km__3.66 km”
See Conical Mountain Climb for solutions.
Modular Arithmetic Puzzle
This is a rather remarkable problem from the 2025 Math Calendar.
“What is the value of
mod33(20252025)
where moda(b) is the remainder r < a after a divides b, that is,
b = ma + r
for some integer m.”
Recall that the answer is a day of the month.
See Modular Arithmetic Puzzle for a solution.
Running in Circles
This is a tricky problem from the Scientific American.
“Miles and Walker begin a workout together at the starting line of a closed-loop track. They run in opposite directions, each at their own constant pace, and continue running until they are both at the starting line at the exact same time again. They pass each other 11 times before this happens (not counting the start or end). Walker is slower than Miles but still manages to complete more than one lap. How many laps does Walker complete?”
I got the wrong answer from just thinking about it, and it took a number of diagrams to see where I had made some false assumptions.
See Running in Circles for solutions.
The B List Puzzle
This is a puzzle from Futility Closet:
“A problem from the Eighth International Mathematical Olympiad, held in Sofia, Bulgaria, in July 1966 (contributed by the Soviet Union):
In a mathematical contest, three problems, A, B, C were posed. Among the participants there were 25 students who solved at least one problem each. Of all the contestants who did not solve problem A, the number who solved B was twice the number who solved C. The number of students who solved only problem A was one more than the number of students who solved A and at least one other problem. Of all students who solved just one problem, half did not solve problem A. How many students solved only problem B?”
See The B List Puzzle for solutions.
Puzzles and Problems: Futility Closet, IMO, algebra
Hex Flex Puzzle
This is another stimulating puzzle from the July 2026 MathsJams Shout, this time by Andy Parkinson.
“What is the area of the hexagon? What is the perimeter?”
See Hex Flex Puzzle for a solution.
Dungeons & Diagrams
This is a fun puzzle from the July 2026 MathsJams Shout.
- Shade squares to create walls and complete the map of the dungeon.
- Printed numbers indicate the number of walls in that row or column.
- Every unshaded square is either a hallway or part of a treasure room.
- Treasure rooms are always 3×3 with a single entrance and a single piece of treasure.
- Hallways are always one square wide. This means that, outside of treasure rooms, there will never be a 2×2 block of unshaded squares.
- Every dead end contains a monster. Every monster is in a dead end.
- All unshaded squares are connected into a single contiguous shape.
- Diagonally adjacent squares are never considered to be adjacent.
Example
These instructions and more puzzles (from zachtronics.com) can be found here. For example, here is a slightly more challenging puzzle.
For an interactive online version see here.
See Dungeons & Diagrams for solutions.
Bicycle Figures Riding
This is a puzzle from Boris Kordemsky’s 1972 Moscow Puzzles.
“Four cyclists do their act on circular paths, each 1/3 mile long. They start simultaneously at the black spots, with speeds of 6, 9, 12, and 15 miles per hour. By the end of the act (20 minutes), how many times will they have simultaneously returned to the spots where they started?”
See Bicycle Figures Riding for solutions.
Area of Rectangle
This is a simple problem from the 2026 Math Calendar.
“Find the area of the rectangle.”
Again, the result must be a number of a day in a month.
See Area of Rectangle for solutions.
This is another healthy workout with logarithms from the 1984 American Invitational Mathematics Exam (AIME).
