This is a belated Christmas puzzle from December 2019 MathsMonday.
“A Christmas tree is made by stacking successively smaller cones. The largest cone has a base of radius 1 unit and a height of 2 units. Each smaller cone has a radius 3/4 of the previous cone and a height 3/4 of the previous cone. Its base overlaps the previous cone, sitting at a height 3/4 of the way up the previous cone.
What are the dimensions of the smallest cone, by volume, that will contain the whole tree for any number of cones?”
Recall that the volume of a cone is π r2 h/3.
See Another Christmas Tree Puzzle
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 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
The craziness of manipulating radicals strikes again. This 2006 four-star problem from Colin Hughes at Maths Challenge is really astonishing, though it takes the right key to unlock it.
“Problem Consider the following sequence:
For which values of [positive integer] n is S(n) rational?”
See Amazing Radical Sum.
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
This is another stimulating math problem from Colin Hughes’s Maths Challenge website (mathschallenge.net).
Find the exact value of the following infinite series:
____________1/2! + 2/3! + 3/4! + 4/5! + …”
See the Unexpected Sum
This is from the UKMT Senior Challenge of 1999.
What is the sum to infinity of the convergent series
See Fibonacci Fandango
This is a challenging problem from Mathematical Quickies (1967).
“Evaluate the infinite product:”
I came up with a motivated solution using some standard techniques from calculus. Mathematical Quickies had a solution that did not employ calculus, but one which I felt used unmotivated tricks. See the Infinite Product Problem.