Category Archives: Math Inquiries

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.

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Serious Series

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 5th year Ontario high school students seeking entrance and scholarships for the second year at a university.

“If sn denotes the sum of the first n natural numbers, find the sum of the infinite series


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

Number of the Beast

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.

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Pairwise Products

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

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