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
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
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
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.
See Meditation on “Is” in Mathematics III – Heliocentrism
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
(Update 3/17/2020, Update 3/21/2020) Continue reading
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
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.
The subtext of this essay might be “word problems,” since the stream of thoughts that led to the za’irajah (zairja) began with a paper I read, while searching for potential problems for this website, on the history of word problems in high school texts in algebra in the 20th and 21st centuries. The following statement by Lorenat caught my attention:
“The newer characteristics of how word problems are treated in Long’s text  … include adding sympathetic commentary about fear of word problems. ‘And of course, there are those dreaded “Word Problems,” but I’ve solved them all for you, so they’re painless.’
[Lorenat continues,] A more extreme example of this is exhibited in the word problem commentary of Michael W. Kelley’s The Idiot’s Complete Guide to Algebra: Second Edition from 2007, in which he describes word problems as ‘a necessary evil of algebra, jammed in there to show you that you can use algebra in “real life.”’ However, Kelley makes no attempt to write ‘real life’ word problems, and criticizes the uselessness of the word problems he does include.”
This essay is an attempt to rebut such negative views of solving word problems by placing the activity in a more favorable historical context.
See the Za’irajah and Mathematics
This is a surprisingly challenging puzzle from the Mathematics 2020 calendar.
“The sketch is of equally spaced railroad ties drawn in a one point perspective. Two of the ties are perceived to the eye to be 25 feet and 20 feet respectively. What is the perceived length x of the third tie?”
Even though the ties are equally-spaced and of equal length in reality, from the point of view of perspective they are successively closer together and diminishing in length. The trick is to figure out what that compression factor is. I had to review my post on the Perspective Map to get some clues.
See the Railroad Tie Problem
The following problem from Five Hundred Mathematical Challenges was a challenge indeed, even though it appeared to be a standard travel puzzle.
“Problem 118. Andy leaves at noon and drives at constant speed back and forth from town A to town B. Bob also leaves at noon, driving at 40 km per hour back and forth from town B to town A on the same highway as Andy. Andy arrives at town B twenty minutes after first passing Bob, whereas Bob arrives at town A forty-five minutes after first passing Andy. At what time do Any and Bob pass each other for the nth time?”
See the Perpetual Meetings Problem