Here is yet another collection of beautiful geometric problems from Catriona Agg (née Shearer). For some reason I found these a bit more challenging than the previous ones. Some of them required more time to “see” the breakthrough.

# Rotating Plane Problem

Here is another challenging problem from the first issue of the 1874 *The* *Analyst*, which also appears in Benjamin Wardhaugh’s book.

“3. If a line make an angle of 40° with a fixed plane, and a plane embracing this line be perpendicular to the fixed plane, how many degrees from its first position must the plane embracing the line revolve in order that it may make an angle of 45° with the fixed plane?

—Communicated by Prof. A. Schuyler, Berea, Ohio.”

Part of the challenge is to construct a diagram of the problem. I used techniques for a solution that were barely in use when this problem was posed in 1874. The contrast between then and now is most revealing.

See the Rotating Plane Problem

# Wine Into Water Problem

Here is a challenging problem from the 1874 *The* *Analyst*.

“A cask containing *a* gallons of wine stands on another containing *a* gallons of water; they are connected by a pipe through which, when open, the wine can escape into the lower cask at the rate of *c* gallons per minute, and through a pipe in the lower cask the mixture can escape at the same rate; also, water can be let in through a pipe on the top of the upper cask at a like rate. If all the pipes be opened at the same instant, how much wine will be in the lower cask at the end of *t* minutes, supposing the fluids to mingle perfectly?

— Communicated by Artemas Martin, Mathematical Editor of *Schoolday *Magazine, Erie, Pennsylvania.”

I found the problem in Benjamin Wardhaugh’s book where he describes *The* *Analyst*:

“Beginning in 1874 and continuing as *Annals of Mathematics* from 1884 onward, *The Analyst* appeared monthly, published in Des Moines, Iowa, and was intended as “a suitable medium of communication between a large class of investigators and students in science, comprising the various grades from the students in our high schools and colleges to the college professor.” It carried a range of mathematical articles, both pure and applied, and a regular series of mathematical problems of varying difficulty: on the whole they seem harder than those in *The Ladies’ Diary* and possibly easier than the *Mathematical Challenges* in the extract after the next. Those given here appeared in the very first issue.”

I tailored my solution after the “Diluted Wine Puzzle”, though this problem was more complicated. Moreover, the final solution must pass from discreet steps to continuous ones.

There is a bonus problem in a later issue:

“19. Referring to Question 4, (No. 1): At what time will the lower cask contain the greatest quantity of wine?

—Communicated by Prof. Geo. R. Perkins.”

See the Wine Into Water Problem

# Equitable Slice Problem

This is another Brainteaser from the *Quantum* math magazine .

“How can a polygonal line BDEFG be drawn in a triangle ABC so that the five triangles obtained have the same area?”

I found this problem rather challenging, especially when I first tried to solve it analytically (using hyperbolas). Eventually I arrived at a procedure that would accomplish the result. The *Quantum* “solution,” however, was tantamount to just saying divide the triangle into triangles of equal area—without providing a method! That is, no solution at all.

See the Equitable Slice Problem

# Nested Polygons Puzzle

These two interesting problems were posed on MEI’s MathsMonday site on 3 February 2020 and 2 March 2020, respectively. MEI and readers posted various approaches, but I used a method suggested by another problem whose origin I no longer recall.

See the Nested Polygons Puzzle

# Train Wreck Puzzle

I thought it might be interesting to explore the mathematics of a common problem with a store-bought HO model train set that contains a collection of straight track segments and fixed-radius curved track segments that form a simple oval. Invariably an initial run of the train has it careening off the track when the train first meets the curved segment after running along the straight track segments.

Why is that? Well of course the train is going too fast. But even if it slows down enough not to fall off the curve, it still jerks unstably and may derail when it first reaches the beginning of the curve. What is going on?

See the Train Wreck Puzzle

# Three Runners Puzzle

Here is another problem from Presh Talwalkar which he says is adapted from India’s Civil Services Exam.

“There are three runners X, Y, and Z. Each runs with a different uniform speed in a 1000 meters race. If X gives Y a start of 50 meters, they will finish the race at the same time. If X gives Z a start of 69 meters, they will finish the race at the same time. Suppose Y and Z are in a [1000 meter] race. How much of a start should Y give to Z so they would finish the race at the same time?”

Even though Talwalkar’s original graphic showed all the runners in a 1000 meter race, it was not immediately clear to me from the wording that the race between Y and Z was also 1000 meters. But that was the case, so I made it explicit.

See the Three Runners Puzzle

# Rock, Paper, Scissors Problem

Here is another problem from the Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos).

“Wiggins grinned at me. ‘You’ve not played Rock Paper Scissors before, Doctor?’

‘Doesn’t ring a bell,’ I told him.

‘Two of you randomly pick one of the three, and shout your choice simultaneously. There are hand gestures, too. If you both get the same, it’s a draw. Otherwise, scissors beats paper, paper beats rock, and rock beats scissors.’

‘So it’s a way of settling an argument,’ I suggested.

‘You were brought up wrong, Doctor,’ Wiggins said gravely. ‘Look, try it this way. I played a series of ten games with Alice earlier. I picked scissors six times, rock three times, and paper once. She picked scissors four times, rock twice, and paper four times. None of our games were drawn.’ He glanced at Holmes, who nodded. ‘So then, Doctor. What was the overall score for the series?’ ”

See the Rock Paper Scissors Problem

# Passion Kiss Problem

This is a somewhat challenging math cryptogram in a slightly different guise from the Canadian Math Society’s magazine, *Crux Mathematicorum*.

“But you can’t make arithmetic out of passion. Passion has no square root.” (Steve Shagan, *City of Angels, *G.P. Putnam’s Sons, New York, 1975, p. 16.)

On the contrary, show that in the decimal system

has a unique solution.

See the Passion Kiss Problem

# Three Equal Circles

Here is a problem from the *Quantum* magazine, only this time from the “Challenges” section (these are expected to be a bit more difficult than the Brainteasers).

“Three circles with the same radius *r* all pass through a point *H*. Prove that the circle passing through the points where the pairs of circles intersect (that is, points *A*, *B*, and *C*) also has the same radius *r*.”

Indeed, I found this quite challenging. It took me several weeks to work out my approach and details.