You’re out after a fresh snow and see bike tracks. Can you tell by the tracks which way the bicycle was going?
Detective Sherlock Holmes and his partner, Dr. Watson, encountered this bike-direction problem in Arthur Conan Doyle’sThe Adventure of the Priory School. Here’s how the detective worked it out.
“This track, as you perceive, was made by a rider who was going from the direction of the school.”
“Or towards it?”
“No, no, my dear Watson. The more deeply sunk impression is, of course, the hind wheel, upon which the weight rests. You perceive several places where it has passed across and obliterated the more shallow mark of the front one. It was undoubtedly heading away from the school. It may or may not be connected with our inquiry, but we will follow it backwards before we go any farther.”
We did so, and at the end of a few hundred yards lost the tracks as we emerged from the boggy portion of the moor. Following the path backwards, we picked out another spot, where a spring trickled across it. Here, once again, was the mark of the bicycle, though nearly obliterated by the hoofs of cows. After that there was no sign, but the path ran right on into Ragged Shaw, the wood which backed on to the school. From this wood the cycle must have emerged. Holmes sat down on a boulder and rested his chin in his hands.
Was Mr. Holmes’s logic correct? What’s an alternative way (there’s more than one) to figure out which way a bicycle went based on its tracks?
We also have a bonus puzzle this week based on a story used to illustrate the perils of short-term thinking in computer programming. The tale is by Joel Spolsky, a software engineer and writer:
Shlemiel gets a job as a street painter, painting the dotted lines down the middle of the road. On the first day he takes a can of paint out to the road and finishes 300 yards of the road. “That’s pretty good!” says his boss, “you’re a fast worker!” and pays him a kopeck.
The next day Shlemiel only gets 150 yards done. “Well, that’s not nearly as good as yesterday, but you’re still a fast worker. One hundred and fifty yards is respectable,” and pays him a kopeck.
The next day Shlemiel paints 30 yards of the road. “Only 30!” shouts his boss. “That’s unacceptable! On the first day you did 10 times that much work! What’s going on?”
“I can’t help it,” says Shlemiel. “Every day I get farther and farther away from the paint can!”
Shlemiel the painter’s algorithm highlights a beginner’s error in designing efficient code, but the numbers aren’t quite right. Let’s say Shlemiel actually did complete 300 yards his first day. How much would he have painted his second and third day?
There’s no single right answer for this one. What are some possibilities?
As always, once you’re able to read comments for this post, use Gary Hewitt’s Enhancer to correctly view formulas and graphics. (Click here for an intro.) And send your favorite puzzles to gary.antonick@NYTimes.com.
That’s it for this week. Check back Friday for solutions, including a bicycle-turning video by mathematical sculptor George Hart.