Monthly Archives: April 2023

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 →

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.

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.

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.