Here is another train puzzle, this time from J. A. H. Hunter’s Entertaining Mathematical Teasers:
“Mike had made the [train] trip many times. ‘That’s the morning express from Tulla we’re passing,’ he said. ‘It left Tulla one hour after we pulled out from Brent, but we’re just 25% faster.’ ‘That’s right, and we’re also passing Cove, two-thirds the distance between Brent and Tulla,’ Martin agreed. ‘So we’re both right on schedule.’ Obviously a couple of train buffs! Assuming constant speeds and no stops, how long would it be before they reached Tulla?”
It is hard to believe a whole year has passed since I started this blog. What is even more surprising is that by February I thought I was about done. I had more or less uploaded the math curiosities and problems I had been thinking about over the years and had presented most of the math essays I had written. There are of course only a finite number of math problems in the world, so I thought I was about done. But much to my surprise I kept finding one more thing that interested me, either an essay or math problem. So here I am. We will have to see what the next year brings.
What to say on this anniversary? I think I will give a retrospective about how the website has been received this past year. This is a challenge, since virtually all my visitors have been silent (which means I haven’t faced criticism, but then I generally haven’t received the necessary correctives either). There are ways to glean some information about visitors and I extracted what I could from the simple plugin I use to count visitors to different posts. I hear that Google analytics provides lots of details, such as the country of origin of a visitor (which would be interesting) and the like, but I have avoided Google and the other social networks as much as possible. After all, I am only an old curmudgeon with old-school notions of privacy.
I have been subverted again by a recent post by Ben Orlin, “Geometry Puzzles for a Winter’s Day,” which is another collection of Catriona Shearer’s geometric puzzles, this time her favorites for the month of November 2019 (which Orlin seems to have named himself). I often visit Orlin’s blog, “Math with Bad Drawings”, so it is hard to kick my addiction to Shearer’s puzzles if he keeps presenting collections. Her production volume is amazing, especially as she is able to maintain the quality that makes her problems so special.
The Stained Glass puzzle generated some discussion about needed constraints to ensure a solution. Essentially, it was agreed to make explicit that the drawing had vertical and horizontal symmetry in the shapes, that is, flipping it horizontally or vertically kept the same shapes, though some of the colors might be swapped.
This is a somewhat elegant problem from the 1987 Discover magazine’s Brain Bogglers by Michael Stueben:
“Each dot in the figure at left represents a factory. On which of the city’s 63 intersections should a warehouse be built to make the total distance between it and all the factors as short as possible? (A much simpler solution than counting and totaling the distances is available.)”
Note that the distance is the taxicab distance I discussed in my article South Dakota Travel Problem rather than the distance along straight lines between the warehouse and factories.
If you will pardon the pun, this is a diabolical problem from the collection Five Hundred Mathematical Challenges.
“Problem 5. Calculate the sum
It has a non-calculus solution, but that involves a bunch of manipulations that were not that evident to me, or at least I doubt if I could have come up with them. I was able to reframe the problem using one of my favorite approaches, power series (or polynomials). The calculations are a bit hairy in any case, but I was impressed that my method worked at all.
“Exactly four minutes after starting to run—when the take-up reel was rotating one and a half times as fast as the projecting reel—the film broke. (The hub diameter of the smaller take-up reel is 8 cm and the hub diameter of the projecting reel is 12 cm.) How many minutes of film remain to be shown?”
This feels like another problem where there is insufficient information to solve it, and that makes it fun and challenging. In fact, I was stumped for a while until I noticed something that was the key to completing the solution.
For a number of years I have collected excerpts that portray mathematical ideas in a literary or philosophical setting. I had occasion to read a few of these on the last day of some math classes I was teaching, since there was no point in introducing a new subject before the final exam.
I thought it might be interesting to present some of these excerpts now. They roughly fall into three categories: logic, infinities (Zeno’s Paradoxes, infinite regress), and permutations.
I was really trying to avoid getting pulled into more addictive geometric challenges from Catriona Shearer (since they can consume your every waking moment), but a recent post by Ben Orlin, “The Tilted Twin (and other delights),” undermined my intent. As Orlin put it, “This is a countdown of her three favorite puzzles from October 2019” and they are vintage Shearer. You should check out Olin’s website since there are “Mild hints in the text; full spoilers in the comments.” He also has some interesting links to other people’s efforts. (Olin did leave out a crucial part of #1, however, which caused me to think the problem under-determined. Checking Catriona Shearer’s Twitter I found the correct statement, which I have used here.)
I have to admit, I personally found the difficulty of these puzzles a bit more challenging than before (unless I am getting rusty) and the difficulty in the order Olin listed. Again, the solutions (I found) are simple but mostly tricky to discover. I solved the problems before looking at Olin’s or others’ solutions.