Here is another problem from the 2020 Math Calendar to stimulate your mind.
Remember that the answers to Math Calendar problems must all be whole numbers representing days of the month.
See New Years Sum for a solution.
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 for solutions.
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 for a solution.
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 for solutions.
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 for solutions.
Meditation on “Is” in Mathematics III – Heliocentrism
Given the aggravating times, I thought I would vent my frustration by ranting on a somewhat nonsensical topic: “The fact that the earth revolves around the sun, rather than the sun around the earth.” This assertion is often used to separate the supposed dunces from the enlightened. It is put on the same level as “the fact that the earth is round (a sphere) and not flat” with the dunces labeled “flat-earthers.”
However, I take umbrage with the use of the word “fact” to conflate these two instances as examples of what “is” or what is “true.” I claim the earth is spherical (more or less: a better approximation is an oblate spheroid) or certainly “curved” rather than “flat.” Whereas the statement that the “sun is at the center of the solar system” is not a fact but an arbitrary convenience.
Coronavirus Mathematics
Given the mathematical nature of this website I feel reluctantly impelled to address the coronavirus pandemic. The mathematics behind the spread of infection is basically the same exponential growth that I discussed in the “Math and Religion” post and has recently been explained by the ever-lucid Grant Sanderson at his 3Blue1Brown website.
What I wish to draw attention to is the series of posts on the coronavirus by Kevin Drum on his website at Mother Jones. I have collected his recent posts comparing the spread of the virus in various countries and added some mathematical commentary of my own, which is the content of this post.
But the bottom line seems to be that in virtually all the countries, including the US, the virus infection is spreading at the Italian rate of doubling every 4 days! The readers of this website are sufficiently numerate to realize the frightening import of that number. If that weren’t enough, Kevin provides additional posts on the results of the modeling at the Imperial College that are truly nerve-wracking for someone such as myself in the most vulnerable cohort. The only blessing so far seems to be that, for once, the children are spared.
At this time, I don’t have the stomach to keep updating the post as new numbers come in. That may change. I could address the catastrophe of having ignorance and incompetence at the helm of the national ship of state, but it is too depressing.
See the Coronavirus Mathematics
(Updates 3/17/2020, 3/21/2020, 4/17/2020, 9/20/2020, 10/1/2020) Continue reading
Hard Geometric Problem
This is another problem from the indefatigable Presh Talwalkar.
_ _____Hard Geometry Problem
“In triangle ABC above, angle A is bisected into two 60° angles. If AD = 100, and AB = 2(AC), what is the length of BC?”
See Hard Geometric Problem for solutions.
(Update 7/18/2020, 7/20/2020) Alternative Solution Continue reading
Geometric Puzzle Munificence
Having fallen under the spell of Catriona Shearer’s geometric puzzles again, I thought I would present the latest group assembled by Ben Orlin, which he dubs “Felt Tip Geometry”, along with a bonus of two more recent ones that caught my fancy as being fine examples of Shearer’s laconic style. Orlin added his own names to the four he assembled and I added names to my two, again ordered from easier to harder.
See Geometric Puzzle Munificence.
(Update 4/16/2020) Ben Orlin has another set of Catriona Shearer puzzles 11 Geometry Puzzles That Drive Mathematicians to Madness which I will leave you to see and enjoy. But I wanted to emphasize some observations he included that I think are spot on. Continue reading