“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)”
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.”
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 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.
Here is another good problem from Five Hundred Mathematical Challenges:
“Problem 100. A hexagon inscribed in a circle has three consecutive sides of length a and three consecutive sides of length b. Determine the radius of the circle.”
This problem made me think of the Putnam Octagon Problem. Again my approach might be considered a bit pedestrian. 500 Math Challenges had a slightly slicker solution.
See the Lop-sided Hexagon Problem
I thought it would be interesting to present a recent entry in the mathematician John Baez’s Diary on some extremes in mathematics from the Bourbaki school, namely, how many symbols it would take to define the number “1.”
I don’t know if the “mathematician” Nicolas Bourbaki holds any significance for students today, but in my time (math graduate school in the 1960s) the Bourbaki approach seemed to permeate everything.
My first exposure to Bourbaki was as a humorous figure described by Paul Halmos in his 1957 article in the Scientific American—the humor being that Bourbaki did not exist. As Halmos wrote:
“One of the legends surrounding the name is that about 25 or 30 years ago first-year students at the Ecole Normale Superieure (where most French mathematicians get their training) were annually exposed to a lecture by a distinguished visitor named Nicolas Bourbaki, who was in fact an amateur actor disguised in a patriarchal beard, and whose lecture was a masterful piece of mathematical double-talk. It is necessary to insert a word of warning about the unreliability of most Bourbaki stories. While the members of this cryptic organization have taken no blood oath of secrecy, most of them are so amused by their own joke that their stories about themselves are intentionally conflicting and apocryphal.”
Nicholas Bourbaki was the pseudonym for a group of French mathematicians who wished to write a treatise which would be, as Halmos stated, “a survey of all mathematics from a sophisticated point of view”.
See the Bourbaki World