# Hard Time Conundrum

This problem comes from the “Problems Drive” section of the Eureka magazine published in 1955 by the Archimedeans at Cambridge University, England.  (“The problems drive is a competition conducted annually by the Archimedeans. Competitors work in pairs and are allowed five minutes per question ….”)

“There are ten times as many seconds remaining in the hour as there are minutes remaining in the day. There are half as many minutes remaining in the day as there will be hours remaining in the week at the end of the day.  What time is it on what day?”

One of the hardest parts of the problem is just being able to translate the statements into mathematical terms.  Solvable in 5 minutes?!!!

See the Hard Time Conundrum for a solution.

# Tethered Triangle Puzzle

This is another problem from MEI’s MathsMonday.

“Two equilateral triangles share a common vertex. Show that the lengths marked a and b are equal for any such arrangement.”

This seems quite amazing at first.  One can picture the small triangle swinging back and forth with red bungee chords tethering its bottom vertices to the bottom vertices of the large triangle.  It would seem remarkable that the lengths of the chords would remain equal to each other throughout.

See the Tethered Triangle Puzzle

# Wandering Epicycle

Here is an intriguing problem from the 2021 Math Calendar.

“If the smaller circle of diameter 7 rotates without slipping within the larger circle, what is the length of the path of P?”

The problem did not state clearly how far the smaller circle should rotate.  Its answer implied it should complete just one full (360°) rotation within the larger circle.

Recall that all the answers are integer days of the month.

See the Wandering Epicycle for a solution.

(Update 1/3/2022)  First, this problem is dealt with in more detail and more expansively on the Mathologer Youtube website by Burkard Polster in his 7 December 2018 post on the “Secrets of the Nothing Grinder” (Figure 1).  A further, deeper discussion of epicycles is given in the Mathologer’s 6 July 2018 post on “Epicycles, complex Fourier and Homer Simpson’s orbit” (Figure 2).  And finally, a panoply of related puzzles is given in the 30 December 2021 Mathologer post “The 3-4-7 miracle. Why is this one not super famous” (Figure 3).

This last post reveals the ambiguity of the idea of “one full (360°) rotation” I disingenuously added to the problem to try to get the answer given in Math Calendar version.

For a complete explanation see the Wandering Epicycle Addendum.

# Existence Proofs II

Futility Closet has another example of an existence proof like their previous one taken from Peter Winkler’s 2021 Mathematical Puzzles (see my post “Existence Proofs”):

“Four bugs live on the four vertices of a regular tetrahedron. One day each bug decides to go for a little walk on the tetrahedron’s surface. After the walk, two of the bugs have returned to their homes, but the other two find that they have switched vertices. Prove that there was some moment when all four bugs lay on the same plane.”

# Polygon Rings

This is a nice geometric problem from the Scottish Mathematical Council (SMC) Senior Mathematical Challenge of 2008.

“Mahti has cut some regular pentagons out of card and is joining them together in a ring. How many pentagons will there be when the ring is complete?

She then decides to join the pentagons with squares which have the same edge length and wants to make a ring as before. Is it possible? If so, determine how many pentagons and squares make up the ring and if not, explain why.”

See the Polygon Rings for solutions.

# Greek-Indian Connection — Alexander’s Legacy

In my post, “Causality, Chance, and Connections,” I have already alluded to one of the biggest mysterious connections that has bedeviled me over the years, namely, the brief suggestion I found in an art book over 50 years ago in the mid-1960s that the human images of Buddha that appeared in statues some four to five hundred years after his life came from the influence of Greek settlers left by Alexander the Great around 300 BC in the Gandhara region of then northwest India (now Pakistan).

I spent decades trying to verify this story.  For some time I have wanted to write an article about what I found.  But it was such a vast and nebulous tale, that I was reluctant to hazard my limited view of the matter.  Nevertheless, I finally could not resist, so here is my sketch of the great Greek Buddha mystery.

See the Greek-Indian Connection

(Update 1/8/2024)  Buddha in Egypt

# Circumscribed House Problem

Here is another problem from the “Brainteasers” section of the Quantum magazine.

“Side AE of pentagon ABCDE equals its diagonal BD. All the other sides of this pentagon are equal to 1. What is the radius of the circle passing through points A, C, and E?”

See the Circumscribed House Problem for solutions.

# MoMath Mazes

I found these mazes on Twitter and thought they might make a relaxing puzzle interlude.  They come from photographs of the street in front of the Museum of Mathematics (MoMath) in New York.   The idea is to traverse the mazes from the Start to the Goal making only right turns.  It was difficult working out the pattern of the green maze, especially the upper right corner.

See the MoMath Mazes

Puzzles and Problems: MoMath