I found an interesting geometric statement in a paper of Glen Van Brummelen cited in the online MAA January 2020 issue of Convergence:
“For instance, Abū’l-Wafā’ describes how to embed an equilateral triangle in a square, as follows: extend the base GD by an equal distance to E. Draw a quarter circle with centre G and radius GB; draw a half circle with centre D and radius DE. The two arcs cross at Z. Then draw an arc with centre E and radius EZ downward, to H. If you draw AT = GH and connect B, H, and T, you will have formed the equilateral triangle.”
So the challenge is to prove this statement regarding yet another fascinating appearance of an equilateral triangle.
See the Triangle of Abū’l-Wafā’
I thought there was nothing new we could learn about Abraham Lincoln, but I see I was quite mistaken after reading Sidney Blumenthal’s article, “Abraham Lincoln, Tech Entrepreneur”.
In the current oppressive anti-science climate it is important to look back at our history and see how integral scientific thinking was to our founding and development. Not only were our Founding Fathers scientists, such as Jefferson and Franklin who with others founded the American Philosophical Society in 1743, but it turns out that President Abraham Lincoln could also lay claim to a scientific mind. Blumenthal’s article describes in detail how Lincoln employed science to advance the development of our country. You should read the entire article, but I am including some highlights.
See Abraham Lincoln Technologist
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
This turned out to be a challenging geometric problem from Poo-Sung Park posted at the Twitter site #GeometryProblem
“Geometry Problem 92: What is the ratio of a:b?”
See the Envelope Puzzle
Here is another problem from the Quantum magazine, only this time from the “Challenges” section (these are expected to be a bit more difficult than the Brainteasers).
“A quadrangle is inscribed in a parallelogram whose area is twice that of the quadrangle. Prove that at least one of the quadrangle’s diagonals is parallel to one of the parallelogram’s sides. (E. Sallinen)”
See Quadrangle in Parallelogram
Here are three counting puzzles from Alex Bellos’s book, Can You Solve My Problems? Bellos recalls the famous legend of the young Gauss in the 19th century who summed up the whole numbers from 1 to 100 by finding a pattern that would simplify the work. Bellos also mentioned that Alcuin some thousand years earlier had discovered a similar, but different, pattern to sum up the numbers. In presenting these three problems he said, “The lesson … is this: If you’re asked to add up a whole bunch of numbers, don’t undertake the challenge literally. Look for the pattern and use it to your advantage.”
See Three Counting Puzzles
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.
Again we have a puzzle from the Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos).
“Our pursuit of the dubious Alan Grey, whom we encountered during The Adventure of the Third Carriage, led Holmes and myself to a circular running track where, as the sun fell, we witnessed a race using bicycles. There was some sort of substantial wager involved in the matter, as I recall, and the track had been closed off specially for the occasion. This was insufficient to prevent our ingress, obviously.
One of the competitors was wearing red, and the other blue. We never did discover their names. As the race started, red immediately pulled ahead. A few moments later, Holmes observed that if they maintained their pace, red would complete a lap in four minutes, whilst blue would complete one in seven.
Having made that pronouncement, he turned to me. ‘How long would it be before red passed blue if they kept those rates up, old chap?’
Whilst I wrestled with the answer, Holmes went back to watching the proceedings. Can you find the solution?”
See the Track Problem
This is a cute little problem I came across via James Tanton (spoiler alert) on Twitter by Ayush DM:
“Here is an old Watsapp problem. How high is the table? Also find the height of the cat and tortoise.”
See Table Tabby Tortoise Problem
This is a nice variation on the typical clock problem posed by Cary Mallon and retweeted by Henk Reuling:
“This clock has fallen on the floor, and unfortunately, there is no indication which way ‘up’ the clock should hang. However, both hands are pointing precisely at the [adjacent] minute marks. You can now work out what the time is.”
See the Fallen Clock Puzzle.