This is another simple problem from the 2025 Math Calendar.
“The sum of x consecutive numbers is 529. Their average is x. What is x?”
For a bonus problem, not included in the calendar, what is the sequence of numbers?
See Number Average for a solution.
This is a puzzle from Futility Closet.
“A puzzle by Steven T., a systems engineer at the National Security Agency, from the NSA’s September 2016 Puzzle Periodical:
Three athletes (and only three athletes) participate in a series of track and field events. Points are awarded for 1st, 2nd, and 3rd place in each event (the same points for each event, i.e. 1st always gets “x” points, 2nd always gets “y” points, 3rd always gets “z” points), with x > y > z > 0, and all point values being integers.
The athletes are named Adam, Bob, and Charlie.
Question: Who finished second in the 100-meter dash (and why)?”
I thought this puzzle impossible at first. There didn’t seem to be enough information to solve it. But a bit of trial-and-error opened a path.
See NSA Track and Field Puzzle for solutions.
This is another puzzle from BL’s Weekly Math Games.
“a + b + c = 2, and
a2 + b2 + c2 = 12
where a, b, and c are real numbers. What is the difference between the maximum and minimum possible values of c?”
The original problem statement mentioned a fourth real number d, but I considered it a typo, since it was not involved in the problem.
See Sphere and Plane Puzzle for a solution.
Presh Talwalkar has an interesting new problem.
“Students and teachers found a recent test in New Zealand to be confusing and challenging for covering topics that were not taught in class.
For the equation below, find the value of k for which the equation has numerically equal but opposite signs (for example, 2 and –2):
”
The problem didn’t mention how old the students were, but the solution to another problem on the test indicates they needed to know calculus.
See Mystery Quadratic for a solution.
This is a nice problem from Five Hundred Mathematical Challenges.
“Problem 62. A plane flies from A to B and back again with a constant engine speed. Turn-around time may be neglected. Will the travel time be more with a wind of constant speed blowing in the direction from A to B than in still air? (Does your intuition agree?)”
See Air Travel for a solution.
This is a 2017 Center of Mathematics Problem of the Week.
“The average age of a 10 member committee is the same as it was 4 years ago, because an older aged member has been replaced by a younger aged member. How much younger is the young member than the older member they replaced?”
See the Young and Old Problem.
This is an interesting algebra problem from BL’s Weekly Math Games, which is behind a subscription wall.
(a3 – b3)/(a – b)3 = 73/3,
what is a – b?”
In fact, it is possible to solve for a and b individually as well.
See a Tricky Ratio Puzzle for the solution.
(Updates 11/6/2024, 11/9/2024) Other Solutions Continue reading
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.
Presh Talwalkar presented an interesting puzzle that originated in the Ladies’ Diary of 1739-40, was recast by Henry Dudeney in 1917, and further modified using American money.
“Each of three Dutchmen, named Hendrick, Elas, and Cornelius has a wife. The three wives have names Gurtrün, Katrün, and Anna (but not necessarily matching the husband’s names in that order). All six go to the market to buy hogs.
Each person buys as many hogs as he or she pays dollars for one. (1 hog costs $1, 2 hogs are $2 each, 3 hogs cost $3 each, etc.) In the end, each husband has spent $63 more than his wife. Hendrick buys 23 more hogs than Katrün, and Elas 11 more than Gurtrün. Now, what is the name of each man’s wife?”
See the Three Dutchmen Puzzle for solutions.
This is a rather mind-boggling problem from the 1947 Eureka magazine.
“Six men, A, B, C, D, E, F, of negligible honesty, met on a perfectly rough day, each carrying a light inextensible umbrella. Each man brought his own umbrella, and took away—let us say “borrowed”—another’s. The umbrella borrowed by A belonged to the borrower of B’s umbrella. The owner of the umbrella borrowed by C borrowed the umbrella belonging to the borrower of D’s umbrella. If the borrower of E’s umbrella was not the owner of that borrowed by F, who borrowed A’s umbrella?”
See the Umbrella Problem for solutions.
