Here is a seemingly simple problem from Futility Closet.
“A quickie from Peter Winkler’s Mathematical Puzzles, 2021: Can West Virginia be inscribed in a square? That is, is it possible to draw some square each of whose four sides is tangent to this shape?”
Technically we might rephrase this as, can we inscribe a flat map of West Virginia in a square, since the boundary of most states is probably not differentiable everywhere, that is, has a tangent everywhere.
But the real significance of the problem is that it is an example of an “existence proof”, which in mathematics refers to a proof that asserts the existence of a solution to a problem, but does not (or cannot) produce the solution itself. These proofs are second in delight only to the “impossible proofs” which prove that something is impossible, such as trisecting an angle solely with ruler and compass.
Here is another classic example (whose origin I don’t recall). Consider the temperatures of the earth around the equator. At any given instant of time there must be at least two antipodal points that have the same temperature. (Antipodal points are the opposite ends of a diameter through the center of the earth.)
See Existence Proofs (revised)
(Update 10/2/2021) I fixed a minor typo: “tail” should have been “head”
Here is a challenging problem from the Polish Mathematical Olympiads published in 1960.
“22. Prove that the polynomial
x44 + x33 + x22 + x11 + 1
is divisible by the polynomial
x4 + x3 + x2 + x + 1.”
See the Polynomial Division Problem
(Update 8/23/2021) The idea expressed in this post that mathematicians are “lazy” and seek short-cuts to solving questions and problems, as I did in this one, was recently the subject of a Numberphile post by Marcus du Sautoy: “Mathematics is all about SHORTCUTS“.
This seemingly magical result from Futility Closet defies proof at first. Go to the Wolfram demo by Jay Warendorff and then …
“Grab point B above and drag it to a new location. Surprisingly, M, the midpoint of RS, doesn’t move.
This works for any triangle — draw squares on two of its sides, note their common vertex, and draw a line that connects the vertices of the respective squares that lie opposite that point. Now changing the location of the common vertex does not change the location of the midpoint of the line.
It was discovered by Dutch mathematician Oene Bottema.”
As we shall see, Bottema’s Theorem has shown up in other guises as well.
See Bottema’s Theorem
Here is surprising problem from the 1875 The Analyst
“81. By G. W. Hill, Nyack Turnpike, N. Y. — Prove that, identically,
By “identically” the proposer means for all n = 1, 2, 3, ….
See the Surprising Identity
(Update 8/20/2021) James Propp at his website has an informative, extensive article on mathematical induction and its variations.
Futility Closet describes a result that is startling, amazing, and mysterious.
“This is pretty: If you choose n > 1 equally spaced points on a unit circle and connect one of them to each of the others, the product of the lengths of these chords equals n.”
The Futility Closet posting includes an interactive display using Wolfram Technology by Jay Warendorff that let’s you select different n and see the results. It also includes a reference to a paper that proves the result; only the paper uses residue theory from complex variables, which seems a bit over-kill, though slick, for such a problem. I found a simpler route.
See a Self-Characterizing Figure
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
Here is a fairly straight-forward problem from 500 Mathematical Challenges.
“Problem 256. Let n be a positive integer. Show that (x – 1)2 is a factor of xn – n(x – 1) – 1.”
See Playing with Polys
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 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
This is another problem from the 2020 Math Calendar.
“Find the difference between the highest and lowest roots of
f(x) = x3 – 54x2 + 969x – 5780”
See Root Difference