Lopsided Hexagon Problem

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

The Bourbaki World

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

Quad in Circle Problem

Here is another Brain Bogglers problem from 1987 by Michael Stueben.

“A quadrilateral with sides three, two, and four units in length is inscribed in a circle of diameter five.  What’s the length of the fourth side of the quadrilateral?”

Like a number of other Brain Bogglers this problem also uses an insight that makes the solution easy. 

See the Quad in Circle Problem.

Red Star (Красная Звезда)

Here is another Brainteaser from the Quantum magazine.

“Prove that the area of the red portion of the star is exactly half the area of the whole star. (N. Avilov)”

This is a relatively simple problem, but I wanted to include it because of its cartoon. Its implied gentle post-Soviet humor reminded me of that strange decade in US-Russian affairs between the end of the Cold War and the rise of Putin in the 21st century. The strangeness was brought home when we had our annual security checks of our classified document storage. Being mostly anti-submarine warfare (ASW) material the main concern was that it would not fall into the hands of the Soviets. But with the “demise” of the Soviet Union in 1989 no one cared any more about the classification. After decades of painfully securing these documents we could not suddenly turn them loose and throw them into the public trash. So we kept them secure anyway. You can imagine how we old cold-warriors feel about the current regime.

That is not to say that I didn’t welcome the thaw. Russian literature, both classical and even “Soviet realism”, as well as Russian cinema, is some of the world’s best. And Russian mathematicians have always been superior, and especially adept at communicating with novices. The collaboration of the American mathematicians and Kvant contributors in Quantum produced excellent results during the thaw. It is unfortunate that it could not survive the rise of Putin and his oligarchs.

See the Red Star

Calculating on the Way

In looking through some old files I came across a math magazine I had bought in 1998. It was called Quantum and was published by the National Science Teachers Association in collaboration with the Russian magazine Kvant during the period 1990 to 2001 (coinciding with the Russian thaw, which in the following age of Putin seems eons ago). Fortunately, they are all online now. Besides some fascinating math articles the magazine contains a column of “Brainteasers.” Here is one of them:

“Alice used to walk to school every morning, and it took 20 minutes for her from door to door. Once on her way she remembered she was going to show the latest issue of Quantum to her classmates but had forgotten it at home. She knew that if she continued walking to school at the same speed, she’d be there 8 minutes before the bell, and if she went back home for the magazine she’d arrive at school 10 minutes late. What fraction of the way to school had she walked at that moment in time? (S. Dvorianinov)”

This is fairly straight-forward, but other problems in the magazine are a bit more challenging.

See Calculating on the Way

Circle in Slot Problem

Here is another UKMT Senior Challenge problem from 2017, which has a straight-forward solution:

“The diagram shows a circle of radius 1 touching three sides of a 2 x 4 rectangle. A diagonal of the rectangle intersects the circle at P and Q, as shown.

What is the length of the chord PQ?

__A_√5____B_4/√5____C_√5 – 2/√5____D_5√5/6____E_2”

See the Circle in Slot Problem

The Bicycle Problem

A fun, relatively new, Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos) has puzzles couched in terms of the Holmes-Watson banter. The following problem is a variation on the Sam Loyd Tandem Bicycle Puzzle.

“ ‘Here’s something mostly unrelated for you to chew over, my dear Watson. Say you and I have a single bicycle between us, and no other transport options save walking. We want to get the both of us to a location eighteen miles distant as swiftly as possible. If my walking speed is five miles per hour compared to your four, but for some reason—perhaps a bad ligament—my cycling speed is eight miles per hour compared to your ten. How would you get us simultaneously to our destination with maximum rapidity?’

‘A cab,’ I suggested.

‘Without cheating,’ Holmes replied, and went back to tossing his toast in the air.”

See the Bicycle Problem

Rubber Band Ant

This is a stimulating little problem from the ever-creative James Tanton:

“An ant is at the east end of an infinite stretchy band, initially 2 ft long. Each day: ant walks 1 ft west on the band. Overnight while sleeping, band stretches to double its length (carrying ant westward as does so). Same routine each day/night. Will ant cover 99% of band’s length?”

(Ant from clipart-library.com)

See the Rubber Band Ant