Category Archives: Puzzles and Problems

Calculating on the Way

In looking through some old files I came across a math magazine I had bought in 1998. It was called Quantum and was published by the National Science Teachers Association in collaboration with the Russian magazine Kvant during the period 1990 to 2001 (coinciding with the Russian thaw, which in the following age of Putin seems eons ago). Fortunately, they are all online now. Besides some fascinating math articles the magazine contains a column of “Brainteasers.” Here is one of them:

“Alice used to walk to school every morning, and it took 20 minutes for her from door to door. Once on her way she remembered she was going to show the latest issue of Quantum to her classmates but had forgotten it at home. She knew that if she continued walking to school at the same speed, she’d be there 8 minutes before the bell, and if she went back home for the magazine she’d arrive at school 10 minutes late. What fraction of the way to school had she walked at that moment in time? (S. Dvorianinov)”

This is fairly straight-forward, but other problems in the magazine are a bit more challenging.

Answer.

See Calculating on the Way for solutions.

Circle in Slot Problem

Here is another UKMT Senior Challenge problem from 2017, which has a straight-forward solution:

“The diagram shows a circle of radius 1 touching three sides of a 2 x 4 rectangle. A diagonal of the rectangle intersects the circle at P and Q, as shown.

What is the length of the chord PQ?

__A_√5____B_4/√5____C_√5 – 2/√5____D_5√5/6____E_2”

Answer.

See the Circle in Slot Problem for a solution.

The Bicycle Problem

A fun, relatively new, Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos) has puzzles couched in terms of the Holmes-Watson banter. The following problem is a variation on the Sam Loyd Tandem Bicycle Puzzle.

“ ‘Here’s something mostly unrelated for you to chew over, my dear Watson. Say you and I have a single bicycle between us, and no other transport options save walking. We want to get the both of us to a location eighteen miles distant as swiftly as possible. If my walking speed is five miles per hour compared to your four, but for some reason—perhaps a bad ligament—my cycling speed is eight miles per hour compared to your ten. How would you get us simultaneously to our destination with maximum rapidity?’

‘A cab,’ I suggested.

‘Without cheating,’ Holmes replied, and went back to tossing his toast in the air.”

Answer.

See the Bicycle Problem for solutions.

Rubber Band Ant

This is a stimulating little problem from the ever-creative James Tanton:

“An ant is at the east end of an infinite stretchy band, initially 2 ft long. Each day: ant walks 1 ft west on the band. Overnight while sleeping, band stretches to double its length (carrying ant westward as does so). Same routine each day/night. Will ant cover 99% of band’s length?”

(Ant from clipart-library.com)

Answer.

See the Rubber Band Ant for solutions.

Hard Geometric Problem

This is another problem from the indefatigable Presh Talwalkar.

_    _____Hard Geometry Problem
“In triangle ABC above, angle A is bisected into two 60° angles. If AD = 100, and AB = 2(AC), what is the length of BC?”

Answer.

See Hard Geometric Problem for solutions.

(Update 7/18/2020, 7/20/2020)  Alternative Solution Continue reading →

Geometric Puzzle Munificence

Having fallen under the spell of Catriona Shearer’s geometric puzzles again, I thought I would present the latest group assembled by Ben Orlin, which he dubs “Felt Tip Geometry”, along with a bonus of two more recent ones that caught my fancy as being fine examples of Shearer’s laconic style. Orlin added his own names to the four he assembled and I added names to my two, again ordered from easier to harder.

See Geometric Puzzle Munificence.

(Update 4/16/2020)  Ben Orlin has another set of Catriona Shearer puzzles 11 Geometry Puzzles That Drive Mathematicians to Madness which I will leave you to see and enjoy. But I wanted to emphasize some observations he included that I think are spot on. Continue reading →

Railroad Tie Problem

This is a surprisingly challenging puzzle from the Mathematics 2020 calendar.

“The sketch is of equally spaced railroad ties drawn in a one point perspective. Two of the ties are perceived to the eye to be 25 feet and 20 feet respectively. What is the perceived length x of the third tie?”

Even though the ties are equally-spaced and of equal length in reality, from the point of view of perspective they are successively closer together and diminishing in length. The trick is to figure out what that compression factor is. I had to review my post on the Perspective Map to get some clues.

See the Railroad Tie Problem

Perpetual Meetings Problem

The following problem from Five Hundred Mathematical Challenges was a challenge indeed, even though it appeared to be a standard travel puzzle.

“Problem 118. Andy leaves at noon and drives at constant speed back and forth from town A to town B. Bob also leaves at noon, driving at 40 km per hour back and forth from town B to town A on the same highway as Andy. Andy arrives at town B twenty minutes after first passing Bob, whereas Bob arrives at town A forty-five minutes after first passing Andy. At what time do Any and Bob pass each other for the nth time?”

Answer.

See the Perpetual Meetings Problem for solutions.

Pinwheel Area Problem

Here is another engaging problem from Presh Talwalkar.

“___________Triangle Area 1984 AIME
Point P is in the interior of triangle ABC, and the lines through P are parallel to the sides of ABC. The three triangles shown in the diagram have areas of 4, 9, and 49. What is the area of triangle ABC?”

Answer.

See the Pinwheel Area Problem. for solutions.

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