Tag Archives: travel puzzles

Tire Wear

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.)

Answer.

See Tire Wear for solutions.

Air Travel

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?)”

Answer

See Air Travel for a solution.

Horses to Qi

This is a challenging 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.

“Now a good horse and an inferior horse set out from Chang’an to Qi.  Qi is 3000 li from Chang’an.  The good horse travels 193 li on the first day and daily increases by 13 li; the inferior horse travels 97 li on the first day and daily decreases by ½ li.  The good horse reaches Qi first and turns back to meet the inferior horse.  Tell: how many days until they meet and how far has each traveled?”

The solution involves common fractions, which the Chinese were already adept at using by 100 BC.

Answer.

See Horses to Qi for a solution.

Locating the Loot

This is a straight-forward problem from Geoffrey Mott-Smith in 1954.

“A brown Terraplane car whizzed past the State Police booth, going 80 miles per hour. The trooper on duty phoned an alert to other stations on the road, then set out on his motorcycle in pursuit. He had gone only a short distance when the brown Terraplane hurtled past him, go­ing in the opposite direction. The car was later caught by a road block, and its occupants proved to be a gang of thieves who had just robbed a jewelry store.

Witnesses testified that the thieves had put their plunder in the car when they fled the scene of the crime. But it was no longer in the car when it was caught. Reports on the wild ride showed that the only time the car could have stopped was in doubling back past the State Police booth.

The trooper reported that the point at which the car passed him on its return was just 2 miles from his booth, and that it reached him just 7 minutes after it had first passed his booth. On both occasions it was apparently making its top speed of 80 miles per hour.

The investigators assumed that the car had made a stop and turned around while some members of the gang cached the loot by the roadside, or perhaps at the office of a “fence.” In an effort to locate the cache, they assumed that the car had maintained a uniform speed, and allowed 2 minutes as the probable loss of time in bringing the car to a halt, turning it, and regaining full speed.

On this assumption, what was the farthest point from the booth that would have to be covered by the search for the loot?”

Answer.

See Locating the Loot for solutions.

Timing the Car

This is yet another simple problem from Henry Dudeney.

“57. TIMING THE CAR

“I was walking along the road at three and a half miles an hour,” said Mr. Pipkins, “when the car dashed past me and only missed me by a few inches.”

“Do you know at what speed it was going?” asked his friend.

“Well, from the moment it passed me to its disappearance round a corner I took twenty-seven steps and walking on reached that corner with one hundred and thirty-five steps more.”

“Then, assuming that you walked, and the car ran, each at a uniform rate, we can easily work out the speed.” ”

Answer.

See Timing the Car for a solution.

Storm Chaser Problem

This is a somewhat challenging problem from the 1997 American Invitational Mathematics Exam (AIME).

“A car travels due east at 2/3 miles per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at √2/2 miles per minute. At time t = 0, the center of the storm is 110 miles due north of the car. At time t = t1 minutes, the car enters the storm circle, and at time t = t2 minutes, the car leaves the storm circle. Find (t1 + t2)/2.”

Answer.

See the Storm Chaser Problem for solutions.

Escalator Puzzle

This is a problem from the 1987 American Invitational Mathematics Exam (AIME).

“Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al’s speed of walking (in steps per unit time) is three times Bob’s walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.)”

Answer.

See the Escalator Puzzle for solutions.

Handicap Racing

This is a nice variation on a racing problem by Geoffrey Mott-Smith from 1954.

“On one side of the playground some of the children were holding foot-races, under a supervisor who handicapped each child according to age and size. In one race, she placed the big boy at the starting line, the little boy a few paces in front of the line, and she gave the little girl twice as much headstart over the little boy as he had over the big boy. The big boy won the race nevertheless. He overtook the little boy in 6 seconds, and the little girl 4 seconds later.

Assuming that all three runners maintained a uniform speed, how long did it take the little boy to overtake the little girl?”

Answer.

See the Handicap Racing for solution.

Close Race Puzzle

This puzzle from the Scottish Mathematical Council (SMC) Senior Mathematics Challenge seems at first to have insufficient information to solve.

“Ant and Dec had a race up a hill and back down by the same route. It was 3 miles from the start to the top of the hill. Ant got there first but was so exhausted that he had to rest for 15 minutes. While he was resting, Dec arrived and went straight back down again. Ant eventually passed Dec on the way down just half a mile before the finish.

Both ran at a steady speed uphill and downhill and, for both of them, their downhill speed was one and a half times faster than their uphill speed. Ant had bet Dec that he would beat him by at least a minute.

Did Ant win his bet?”

Answer.

See the Close Race Puzzle for solutions.

(Update 1/2/2023)  Alternative Solution from Oscar Rojas Continue reading →

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