Here is another sum problem, this time from the 2021 Math Calendar.

________________

As before, recall that all the answers are integer days of the month. And the solution employs a technique familiar to these pages.

See the Winter Sum

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Here is another sum problem, this time from the 2021 Math Calendar.

________________

As before, recall that all the answers are integer days of the month. And the solution employs a technique familiar to these pages.

See the Winter Sum

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

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

Here is another problem from the 2020 Math Calendar.

As a hint, recall that all the answers are integer days of the month. And the solution employs a technique familiar to these pages.

See Autumn Sum

The following problem comes from a 1961 exam set collected by Ed Barbeau of the University of Toronto. The discontinued exams (by 2003) were for 5^{th} year Ontario high school students seeking entrance and scholarships for the second year at a university.

“If *s _{n}* denotes the sum of the first

.”

Unfortunately, the “Grade XIII” exam problem sets were not provided with answers, so I have no confirmation for my result. There may be a cunning way to manipulate the series to get a solution, but I could not see it off-hand. So I employed my tried and true power series approach to get my answer. It turned out to be power series manipulations on steroids, so there must be a simpler solution that does not use calculus. I assume the exams were timed exams, so I am not sure how a harried student could come up with a quick solution. I would appreciate any insights into this.

See Serious Series

(**Update 1/18/2021**) Another Solution Continue reading

This is a delightful and surprising problem from Presh Talwalkar.

“This puzzle was created by a MindYourDecisions fan in India. What is the value of the infinite product? The numerators are the odd nth roots of [Euler’s constant] e and the denominators are even nth roots of e.”

See Euler Magic

If you will pardon the pun, this is a diabolical problem from the collection *Five Hundred Mathematical Challenges*.

“**Problem 5**. Calculate the sum

__________

It has a non-calculus solution, but that involves a bunch of manipulations that were not that evident to me, or at least I doubt if I could have come up with them. I was able to reframe the problem using one of my favorite approaches, power series (or polynomials). The calculations are a bit hairy in any case, but I was impressed that my method worked at all.

See the Number of the Beast

Here is another Brain Bogglers problem from 1987.

“Exactly four minutes after starting to run—when the take-up reel was rotating one and a half times as fast as the projecting reel—the film broke. (The hub diameter of the smaller take-up reel is 8 cm and the hub diameter of the projecting reel is 12 cm.) How many minutes of film remain to be shown?”

This feels like another problem where there is insufficient information to solve it, and that makes it fun and challenging. In fact, I was stumped for a while until I noticed something that was the key to completing the solution.

See the Movie Projector Problem.

This 2005 four-star problem from Colin Hughes at *Maths Challenge* is also a bit challenging.

“**Problem**

For any set of real numbers, R = {x, y, z}, let sum of pairwise products,

________________S = xy + xz + yz.

Given that x + y + z = 1, prove that S ≤ 1/3.”

Again, I took a different approach from Maths Challenge, whose solution began with an unexplained premise.

See the Pairwise Products

This 2007 four-star problem from Colin Hughes at *Maths Challenge* is definitely a bit challenging.

“**Problem**

For any positive integer, k, let Sk = {x1, x2, … , xn} be the set of [non-negative] real numbers for which x1 + x2 + … + xn = k and P = x1 x2 … xn is maximised. For example, when k = 10, the set {2, 3, 5} would give P = 30 and the set {2.2, 2.4, 2.5, 2.9} would give P = 38.25. In fact, S10 = {2.5, 2.5, 2.5, 2.5}, for which P = 39.0625.

Prove that P is maximised when all the elements of S are equal in value and rational.”

I took a different approach from Maths Challenge, but for me, it did not rely on remembering a somewhat obscure formula. (I don’t remember formulas well at my age—only procedures, processes, or proofs, which is ironic, since at a younger age it was just the opposite.) It is also clear from the Maths Challenge solution that the numbers were assumed to be non-negative.

See Maximum Product.