After Five O’clock

This is a fairly extensive clock problem by Geoffrey Mott-Smith from 1954.

“The clock shown in the illustration has just struck five. A number of things are going to happen in this next hour, and I am curious to know the exact times.

  1. At what time will the two hands coincide?
  2. At what time will the two hands first stand at right angles to each other?
  3. At one point the hands will stand at an angle of 30 degrees, the minute hand being before the hour hand. Then the former will pass the latter and presently make an angle of 60 degrees on the other side. How much time will elapse between these two events?”

Answer.

See After Five O’Clock for solutions.

Language Students Puzzle

This is a slightly challenging problem from Dan Griller.

“Every pupil at the Euler Academy studies French or Spanish.  At the start of the year, one third of the French students also studied Spanish, and 2 fifths of the Spanish students also studied French.  After one term, six of the double-linguists dropped French, so that now only a quarter of the French students study Spanish.  How many pupils are at the Euler Academy?”

Just to be clear, “French students” means Euler Academy pupils studying French, and similarly for “Spanish students.”

Answer.

See the Language Students Puzzle for solution.

Pythagorean Parabola Puzzle

Since the changes in Twitter (now X), I have not been able to see the posts, not being a subscriber.  But I noticed poking around that some twitter accounts were still viewable.  However, like some demented aging octogenarian they had lost track of time, that is, instead of being sorted with the most recent post first, they showed a random scattering of posts from different times.  So a current post could be right next to one several years ago.  That is what I discovered with the now defunct MathsMonday site.  I found a post from 10 May 2021 that I had not seen before, namely,

“The points A and B are on the curve y = x2 such that AOB is a right angle.  What points A and B will give the smallest possible area for the triangle AOB?”

Answer.

See the Pythagorean Parabola Puzzle for solution.

(Update 9/1/2023) Elegant Alternative Solution by Oscar Rojas
Continue reading

Voice Stealing

This essay is slightly tangential to my usual fare, but it is prompted by a most amazing video that convinced me that the impact of AI this time is not hype, but rather a real threat to our society.  I found the video at 3 Quarks Daily and it was of Johnny Cash singing a song called “Barbie Girl” to the tune of his trademark Folsom Prison Blues—only it wasn’t the late Johnny Cash (1932–2003), it was AI!

See Voice Stealing

(Update 11/2/2023)  I was wondering why the seeming lack of interest in this post, and then I tried the link to the video and found it has been removed from the public.  There must be a reason, probably copyright issues somewhere, so even though I got a copy when it was public, I don’t think I should post it.  This is really too bad, since it is in incredible example of what may be our dystopian future.  I just rooted around and found another link that seems to be working.  I have updated the text above.

(Update 5/4/2024)  It gets worse.  The Atlantic article “My Journey Inside the Voice-Clone Factory” shows what happens when voice-cloning is combined with image-cloning and OpenAI language models.  It is a dystopian nightmare beyond imagining.

Floating Square Puzzle

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

“A mysterious square has materialized in the middle of the MCG, hovering in mid-air. The heights above the ground of three of its corners are 13, 21 and 34 metres. The fourth corner is higher still. How high?”

Answer.

See the Floating Square Puzzle for solutions.

(Update 8/13/2023)  Alternative Solution Continue reading

Square In A Quarter Circle

Another puzzle by Presh Talwalkar.

“Thanks to John H. for the suggestion!

A square is inscribed in a quarter circle such that the outer vertices are on the arc of the quarter circle. If the quarter circle has a radius equal to 1, what is the area of the square?

I am told this was given to 7th grade students (ages 12-13), and I think it is a very challenging problem for that age group. In fact I think it is a good problem for any geometry student.”

Answer.

See the Square in Quarter Circle for solutions.

The Lure of Mathematics Conundrum

A prevalent theme of much of popular mathematical exposition and debates about mathematics education concerns how to interest a wider population in matters mathematical.  For the most part I feel that essays that try to present the “beauty” of mathematics are doomed to failure, as are most discussions of esthetics.  The underlying goal of such writing is a legitimate and laudable attempt to show the appeal of math.  But I fear it succeeds only with those already converted.  So is there another way?

See the Lure of Mathematics Conundrum

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.