This is a puzzle from Boris Kordemsky’s 1972 Moscow Puzzles.
“Two freight trains, each 1/6 mile long and traveling 60 miles per hour, meet and pass each other. How many seconds is it between when the locomotives pass each other and the cabooses pass each other?”
See Trains Meeting for solutions.
This is another problem from Dan Griller.
“When Anthony and Benjamin run round a circular track in the same direction at constant but different speeds, they meet every 3 minutes. When Benjamin changes direction (but maintains his speed) they meet every 40 seconds.
If Anthony is faster than Benjamin, calculate
(speed of Anthony) / (speed of Benjamin)”
See Yet Another Track Puzzle for a solution.
This is another typical travel puzzle from the 2024 Math Calendar.
“A boat travels downriver at 30 mph, then goes back up along the same path at 20 mph. What is the boat’s average speed?”
As before, recall that all the answers are integer days of the month.
See Another Boat Puzzle for a solution.
This is a simple 1917 puzzle from Henry Dudeney.
“During a visit to the seaside Tommy and Evangeline insisted on having a donkey race over the mile course on the sands. Mr. Dobson and some of his friends whom he had met on the beach acted as judges, but, as the donkeys were familiar acquaintances and declined to part company the whole way, a dead heat was unavoidable. However, the judges, being stationed at different points on the course, which was marked off in quarter-miles, noted the following results:—The first three-quarters were run in six and three-quarter minutes, the first half-mile took the same time as the second half, and the third quarter was run in exactly the same time as the last quarter. From these results Mr. Dobson amused himself in discovering just how long it took those two donkeys to run the whole mile. Can you give the answer?
See Donkey Riding for solutions.
This is another problem from the c.100AD Chinese mathematical work, Jiǔ zhāng suàn shù (The Nine Chapters on the Mathematical Art) found at the MAA Convergence website Convergence.
“A square walled city measures 10 li on each side. At the center of each side is a gate. Two persons start walking from the center of the city. One walks out the south gate, the other the east gate. The person walking south proceeds an unknown number of pu then turns northeast and continues past the corner of the city until they meet the eastward traveler. The ratio of the speeds for the southward and eastward travelers is 5:3. How many pu did each walk before they met? [1 li = 300 pu]”
See Two Men Meet for a solution.
This is an interesting problem from the 1966 Eureka magazine.
“A railway and a road run together for seven miles from P to Q. Two miles from P there is a level crossing, which is closed one minute before, and opened one minute after, a train passes.
A train passes a Stationary car at P and travels on to Q at 60 m.p.h., and, forgetting to slow down, crashes at Q; the car passes the train as it crashes. Assuming that stopping for an instant from full speed loses the car one minute, of what speed must it be capable?”
See the Railway Crossing Problem for a solution.
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 take on the passing train type puzzle from the Moscow Puzzles.
“A train moving 45 miles per hour meets and is passed by a train moving 36 miles per hour. A passenger in the first train sees the second train take 6 seconds to pass him. How long is the second train?”
See Another Passing Train Puzzle for solutions.
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.
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.
