Ant Connection Problem

This is a nice puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings.

“Two ants are on a cylindrical glass that is 5 centimetres in diameter. The ants are on opposite sides of the glass, 5 centimetres down from the glass’s rim. If both ants are on the outside of the glass, what is the shortest distance required for one ant to crawl to the other? What if one ant is on the outside of the glass and the other is on the inside?”

Answer.

See the Ant Connection Problem for solution.

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Putnam Ellipse Areas Problem

This is a nifty problem from Presh Talwakar.

“This is adapted from the 1994 Putnam, A2. Thanks to Nirman for the suggestion!

Let R be the region in the first quadrant bounded by the x-axis, the line y = x/2, and the ellipse x2/9 + y2 = 1. Let R‘ be the region in the first quadrant bounded by the y-axis, the line y = mx and the ellipse. Find the value of m such that R and R‘ have the same area.”

Answer.

See the Putnam Ellipse Areas Problem for solution.

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A New Day

One of the physics blogs I enjoy reading is by the mathematical physicist Peter Woit, called Not Even Wrong.  A recent post provided a tantalizing teaser:

“I want to [link to] an insightful explanation of the history of string theory, discussing the implications of how it was sold to the public. It’s by a wonderful young physicist I had never heard of before, Angela Collier. She has a Youtube channel, and her latest video is string theory lied to us and now science communication is hard.

… It’s as hilarious as it is brilliant, and you have to see for yourself.”

Collier delivered her talk lucidly and thoroughly—all while playing a frenetic video game!  She claimed she used the length of the game to time her talk.  Of course we can walk and talk, and ride bicycles and talk, but I have never seen anyone split their mental concentration between a fast-paced video game and an esoteric physics explanation of the history of string theory and supersymmetry—for over 50 minutes!  And there was something about her presentation that was completely captivating.  It was definitely a serious scientific talk, but the ludicrousness of the game-playing echoed how ridiculous the continued, misplaced fascination with string theory is.  Naturally I had to learn more about this provocative physicist.

See A New Day

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Curious Sunbeam Problem

This is a Catriona Agg problem presented by itself, since it turned out to be the most challenging one I ever tried.  Usually I can solve her problems in a few minutes or maybe hours, or sometimes days if they are especially challenging.  But this problem has taken me weeks and I had to rely on a non-geometric argument.  The problem is full of fascinating and unexpected relationships, but I couldn’t find a way to use them to prove the answer.

See the Curious Sunbeam Problem

(Update 5/5/2023, 7/22/2025)  Alternative Solutions Continue reading →

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The Tired Messenger Problem

Here is another challenging problem from the Polish Mathematical Olympiads.  Its generality will cause more thought than for a simpler, specific problem.

“A cyclist sets off from point O and rides with constant velocity v along a rectilinear highway.  A messenger, who is at a distance a from point O and at a distance b from the highway, wants to deliver a letter to the cyclist.  What is the minimum velocity with which the messenger should run in order to attain his objective?”

See the Tired Messenger Problem

(Update 1/29/2025)  Dan Steinitz Solution

Dan Steinitz from Israel has sent an elegant solution that only involves vectors and geometry without calculus.  I have edited slightly his email and added excerpts from his whiteboard solution, though without the Hebrew annotations, which unfortunately I cannot read.  But that is the glory of the universal language of mathematics: it can be read and understood in any language.

See Dan Steinitz Solution.

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Bailing Water Problem

This is a straight-forward problem from Five Hundred Mathematical Challenges.

“A boat has sprung a leak.  Water is coming in at a uniform rate and some has already accumulated when the leak is detected.  At this point, 12 men of equal skill can pump the boat dry in 3 hours, while 5 men require 10 hours.  How many men are needed to pump it dry in 2 hours?”

Answer.

See the Bailing Water Problem for solution.

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Pillar Wrapping Problem

This is a fun problem from the 1949 Eureka magazine.

“The following problems were set at the Archimedeans’ 1949 Problems Drive. Competitors were allowed five minutes for each question.  [This is problem #9.]

A pillar is in the form of a truncated right circular cone. The diameter at the top is 1 ft., at the bottom it is 2 ft. The slant height is 15 ft. A streamer is wound exactly five times round the pillar starting at the top and ending at the bottom. What is the shortest length the streamer can have?”

Answer.

See the Pillar Wrapping Problem for solution.

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Moon Quarters Problem

This is a straight-forward problem from the Scottish Mathematical Council (SMC) Senior Mathematics Challenge.

“A circle has radius 1 cm and AB is a diameter.  Two circular arcs of equal radius are drawn with centres A and B.  These arcs meet on the circle as shown.  Calculate the shaded area.”

There are several possible approaches and the SMC offers two examples.

Answer.

See the Moon Quarters Problem for solution.

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Max Angle Puzzle

Here is a familiar puzzle from the Mathigon Puzzle Calendars for 2021.

“Given a line and two points A and B, which point P on the line forms the largest angle APB?”

See the Max Angle Puzzle

An excellent application of the solution to this puzzle can be found at Numberphile, where Ben Sparks explains an optimal rugby goal-kicking strategy.

(Update 3/23/2023)  Solution Construction

Continue reading →

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