Maximized Box Problem

This problem is from Colin Hughes’s Maths Challenge website (mathschallenge.net).

“Four corners measuring x by x are removed from a sheet of material that measures a by a to make a square based open-top box.  Prove that the volume of the box is maximised iff the area of the base is equal to the area of the four sides.”

See the Maximized Box Problem

Two More Jugs

Here is another classic example of the three jug problem posed in the Mathigon Puzzle Calendars for 2017.

“How can I measure exactly 8 liters of water, using just one 11 liter and one 6 liter bucket?”

It is assumed you have unlimited access to water (the “third jug” of at least 17 liters).  You can only fill or empty the jugs, unless in poring from one jug to another you fill the receiving jug before emptying the poring jug.  (Hint: see the Three Jugs Problem.)

See Two More Jugs.

Elliptical Medians Problem

This is a tantalizing problem from the 1977 Crux Mathematicorum.

“278. Proposed by W.A. McWorter, Jr., The Ohio State University.

If each of the medians of a triangle is extended beyond the sides of the triangle to 4/3 its length, show that the three new points formed and the vertices of the triangle all lie on an ellipse.”

See the Elliptical Medians Problem

Storm Chaser Problem

This is a somewhat challenging problem from the 1997 American Invitational Mathematics Exam (AIME).

“A car travels due east at 2/3 miles per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at √2/2 miles per minute. At time t = 0, the center of the storm is 110 miles due north of the car. At time t = t1 minutes, the car enters the storm circle, and at time t = t2 minutes, the car leaves the storm circle. Find (t1 + t2)/2.”

See the Storm Chaser Problem for solutions.

String of Beads Puzzle

This is a nifty problem from Presh Talwalkar.

“This is from a Manga called Q.E.D. I thank Sparky from the Philippines for the suggestion!

A string of beads is formed from 25 circles of the same size. The string passes through the center of each circle. The area enclosed by the string inside each circle is shaded in blue, and the remaining areas of the circles are shaded in orange. What is the value of the orange area minus the blue area? Calculate the area in terms of r, the radius of each circle.”

See the String of Beads Puzzle for solutions.

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?”

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.”

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?”

See the Pythagorean Parabola Puzzle for solution.

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

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?”

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.”

See the Square in Quarter Circle for solutions.