One time, in a faraway country, there was a champion swimmer who could swim circles around his competitors. He wanted to compete in the world championship of circle swimming, but due to a pandemic he was forced to practice alone. His country's swimming federation built a perfect circular lake in the middle a forest so he could practice safely. To make it easy for him to find his way, workers marked the bottom with concentric circles. Circle swimming, an aquatic technique that allows an athlete to swim faster on curves than they can on straight paths, is amazing. The lake's size was precisely calibrated to allow the fastest swimmer to swim 1 unit over 1 time unit. He had to swim a radius of 3.5 inches in the lake. This meant that he needed to take 3.5 units to complete one circle. A buoy marked the lake's center to aid in his orientation.The forest was not entirely safe, however. While practicing near the lake's center, the swimmer saw a bear looking at him hungry. To assess the situation, he quickly swam to its center. He was sure that the bear was watching him. The bear kept circling the lake at a steady speed and showed no sign of giving up. The bear didn't venture into the lake where the swimmer could easily surpass it. He noticed that even though the bear was slower than he could swim, his running speed was still 3.5x faster than he could swim. He was feeling tired and knew he couldn't stay in the water for too long. He felt confident that if the bear did not catch him, he would be able to quickly sprint into the trees and escape the bear.Before you read on or attempt to do detailed calculations, think about the above scenario. What pattern should the athlete swim to maximize his chances of escape?Do you have a plan? Okay, you are now ready to read.The common gray squirrel is an animal that finds itself in a similar situation to the swimmer. It has developed a strategy to frustrate its pursuers. The squirrel that is being pursued by a dog will always position itself at an opposite point to its pursuer, while simultaneously climbing higher into a tree. It soon makes its escape. As it moves at a faster speed, the squirrel is able to navigate a narrower circle than the dog. This allows the dog to avoid the tree. It is so efficient that an astronaut can use it to escape an enemy space cruiser in Arthur C. Clarkes Hide and Seek.This strategy could be useful in the swimmer situation. To fully solve the problem, detailed calculations are required so let's take a look at a few bite-sized questions. These questions will help you to assume that the bear will naturally follow the strategy that is most beneficial to it.Puzzle 1How does the swimmer use the squirrel strategy to avoid the bear? How many turns does the swimmer make in this manner? How many complete turns will the swimmer make before they stop using the squirrel strategy? What is the time it takes to get there? Is it possible for the swimmer to escape the bear?This is a classic problem, which has been solved before. This problem may seem familiar to some readers. However, we have changed the numbers. You've made it this far? Then, you can try the next set, which I don't believe has been asked before. To find the answer, you can either model the situation or use numerical simulations or analytical techniques.Puzzle 2Let's say our goal is to not only evade the bear, but also to flee as quickly as possible. After all, swimmers legs and arms are tired. Which strategy is the most efficient and which is the fastest to escape in each case?Continue to follow the squirrel strategy until you are unable to do so, then move in the opposite direction. Continue to follow the squirrel strategy until you are unable to do so, then move in another direction. For a while, follow the squirrel strategy and then move in a different direction. You can also follow a different strategy to the squirrel strategy.Puzzle 3On the other hand, suppose that the athlete's goal is to get as far as possible ahead of the bear in the lake. Which strategy is most effective now? And what distance can he put between himself and bear around the lake's circumference?Continue to follow the squirrel strategy until you are unable to do so, then move in the opposite direction. Continue to follow the squirrel strategy until you are unable to do so, then move in another direction. For a while, follow the squirrel strategy and then move in a different direction. You can also follow a different strategy to the squirrel strategy.For those who can't get enough of the puzzle, here are some bonus questions.1 BonusWhat if the lake radius is 4.5 inches and the swimmer's speed is 4.5? Does this affect the best strategy to solve puzzles 2 or 3? (The swimmer's speed is the same as before.Bonus 2What is the maximum ratio between the swimmers' speed and the bear's running speed? (Let's assume that the radius of a lake in units equals this ratio and that the swimmers' speed is the same.This is it for the Insights Puzzle of the Month. I'm eager to hear your solutions to these questions.Be happy to figure, and don't go around in circles!Editor's Note: The Quanta Magazine T-shirt and one of the Quanta books Alice and Bob Meet the Wall of Fire, or The Prime Number Conspiracy (winners option) will be given to the reader who submits the best, most creative, or the most insightful solution. If you would like to suggest a puzzle for an Insights column in the future, please leave a comment below.
