And so another year has passed—a pretty horrible one at that. Hopefully things mathematical have provided a distraction and entertainment.
I thought I would update the retrospective from a year ago. Clearly, this eventful year made itself felt even through the statistics I gathered from the website. The growth the site experienced in posts read hit a plateau, but strangely, the number of new visitors increased a bit, at least for a while. That may have been due to the greater amount of free time away from the classroom that the virus demanded.
See Two Year Anniversary
Here is yet another problem from Presh Talwalkar. This one is rather elegant in its simplicity of statement and answer.
“Solve For The Angle – Viral Puzzle
I thank Barry and also Akshay Dhivare from India for suggesting this problem! This puzzle is popular on social media. What is the measure of the angle denoted by a “?” in the following diagram? You have to solve it using elementary geometry (no trigonometry or other methods). It’s harder than it looks. I admit I did not solve it. Can you figure it out?”
See the Shy Angle Problem.
Here is another delightful problem from the Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos).
“In Sussex, Holmes and I ran into a pair of woodcutters named Doug and Dave. There was an air of the unreliable about them—not helped by a clearly discernable aroma of scrumpy—but they nevertheless proved extremely helpful in guiding us to a particular hilltop clearing some distance outside of the town of Arundel. A shadowy group had been counterfeiting sorceries of a positively medieval kind, and all sorts of nastiness had ensued.
The Adventure of the Black Alchemist is not one that I would feel comfortable recounting, and if my life never drags me back to Chanctonbury Ring I shall be a happy man. But there is still some instructive material here. Whilst we were ascending our hill, Doug and Dave made conversation by telling us about their trade. According to these worthies, working together they were able to saw 600 cubic feet of wood into large logs over the course of a day, or split as much as 900 cubic feet of logs into chunks of firewood.
Holmes immediately suggested that they saw as much wood in the first part of the day as they would need in order to finish splitting it at the end of the day. It naturally fell to me to calculate precisely how much wood that would be.
Can you find the answer?”
See the Loggers Problem
This is a nifty little problem from the Quantum math magazine.
“Two ants stand at opposite corners of a 1-meter square. A barrier was placed between them in the form of half a 1-meter square attached along the diagonal of the first square, as shown in the picture. One ant wants to walk to the other. How long is the shortest path?”
See the Barrier Minimal Path Problem
This is a fun logic puzzle from one of Ian Stewart’s many math collections. I discovered that the problem actually is basically one of Lewis Carroll’s examples from an 1896 book:
- The only animals in this house are cats.
- Every animal that loves to gaze at the moon is suitable for a pet.
- When I detest an animal, I avoid it.
- No animals are meat-eaters, unless they prowl by night.
- No cat fails to kill mice.
- No Animals ever take to me, except those in this house.
- Kangaroos are not suitable for pets.
- Only meat-eaters kill mice.
- I detest animals that do not take to me.
- Animals that prowl at night love to gaze at the moon.
If all these statements are correct, do I avoid kangaroos, or not?
See Do I Avoid Kangaroos?
Here is another collection of beautiful geometric problems from Catriona Agg (née Shearer). They never fail to brighten the day with their loveliness.
See Geometric Puzzle Marvels
Here is a simple Futility Closet problem from 2014.
“This unit square is divided into four regions by a diagonal and a line that connects a vertex to the midpoint of an opposite side. What are the areas of the four regions?”
See the Square Deal
One of my favorite bloggers, Kevin Drum, decided to relieve the tedium of our current political anarchy by whacking the hornets’ nest of the high school mathematics curriculum, in particular the subject of plane geometry. You can tell from the tag list on my blog that I hold plane geometry in high regard and can’t let this gibe pass without some rebuttal, futile as it may be. Actually, I am not going to weigh in on the general issue of the current math curriculum that much, but rather make a few observations from my own experience over the years as it relates to Kevin’s post.
See Down With Geometry
(Update 2/9/2021) Vindication! Continue reading
Here is a problem from Five Hundred Mathematical Challenges that I indeed found quite challenging.
“Problem 235. Two fixed points A and B and a moving point M are taken on the circumference of a circle. On the extension of the line segment AM a point N is taken, outside the circle, so that lengths MN = MB. Find the locus of N.”
Since one of the first hurdles I faced with this problem was trying to figure out what type of shape was being generated, I thought I would omit my usual drawings illustrating the problem statement. There turned out to be a lot of cases to consider, but the result was most satisfying. I also included the case when N is inside the circle. Again Visio was my main tool to handle all the examples with the concomitant requirement to prove whatever Visio suggested.
See the Curve Making Puzzle
This is a nice puzzle from Clifford Pickover in the 1996 Discover magazine’s Brain Bogglers.
“Thoth, ancient Egyptian god of wisdom and learning, has abducted Ahmes, a famous Egyptian scribe, in order to assess his intellectual prowess. Thoth places Ahmes before a large funnel set in the ground. It has a circular opening 1,000 feet in diameter, and its walls are quite slippery. If Ahmes attempts to enter the funnel, he will slip down the wall. At the bottom of the funnel is a sleep-inducing liquid that will instantly put Ahmes to sleep for eight hours if he touches it.
As shown in the illustration, there are two ankh-shaped towers. One stands on a cylindrical platform in the center of the funnel. The platform’s surface is at ground level. The distance from the platform’s surface to the liquid is 500 feet. The other ankh tower is on land, at the edge of the funnel.
Thoth hands Ahmes two objects: a rope 1,006.28 feet in length and the skull of a chicken. Thoth says to Ahmes, ‘If you are able to get to the central tower and touch it, we will live in harmony for the next millennium. If not, I will detain you for further testing. Please note that with each passing hour, I will decrease the rope’s length by a foot.’
How can Ahmes reach the central ankh tower and touch it? ”
See the Thoth Maneuver