Can You Squeeze the Heart?

Happy Valentine’s Day from The Fiddler! This week, you’ll be “squeezing” (i.e., inscribing) heart shapes inside circles.

Welcome to Fiddler on the Proof! The Fiddler is the spiritual successor to FiveThirtyEight’s The Riddler column, which ran for eight years under the stewardship of myself and Ollie Roeder.

Each week, I present mathematical puzzles intended to both challenge and delight you. Puzzles come out Friday mornings (8 a.m. Eastern time). Most can be solved with careful thought, pencil and paper, and the aid of a calculator. Many include “Extra Credit,” where the analysis gets particularly hairy or where you might turn to a computer for assistance.

I’ll also give a shoutout to 🎻 one lucky winner 🎻 of the previous week’s puzzle, chosen randomly from among those who submit their solution before 11:59 p.m. the Monday after that puzzle was released. I’ll do my best to read through all the submissions and give additional shoutouts to creative approaches or awesome visualizations, the latter of which could receive 🎬 Best Picture Awards 🎬.

This Week’s Fiddler

You can generate a heart shape by drawing a unit square (i.e., a square with side length 1), and then attaching semicircles (each with radius 1/2) to adjacent edges, as shown in the diagram below:

What is the radius of the smallest circle that contains this heart shape?

This Week’s Extra Credit

Instead of containing one heart shape, now your circle must contain two heart shapes. Again, each heart consists of a unit square and two semicircular lobes. The two hearts are not allowed to overlap.

What is the radius of the smallest circle that contains these two hearts?

Making the ⌊Rounds⌉

There’s so much more puzzling goodness out there, I’d be remiss if I didn’t share some of it here. This week, I’m sharing a Pythagorean puzzle I came across, which I’ll paraphrase spoiler-free:

Prove that there are infinitely many points on the unit circle whose coordinates are terminating decimals.

Examples include (0.6, 0.8) and (0.28, 0.96).

Feel free to discuss this puzzle and your approach in the comments below!

Want to Submit a Puzzle Idea?

Then do it! Your puzzle could be the highlight of everyone’s weekend. If you have a puzzle idea, shoot me an email. I love it when ideas also come with solutions, but that’s not a requirement.

Standings

I’m tracking submissions from paid subscribers and compiling a leaderboard, which I’ll reset every quarter. All timely correct solutions to Fiddlers and Extra Credits are worth 1 point each. At the end of each quarter, I’ll 👑 crown 👑 the finest of Fiddlers. (If you think you see a mistake in the standings, kindly let me know.)

Last Week’s Fiddler

Congratulations to the (randomly selected) winner from last week: 🎻 D Smith 🎻 from Twente, Netherlands. I received 44 timely submissions, of which 39 were correct—good for an 89 percent solve rate.

Last week, inspired by recent LLM benchmarking experiments, you had a unit square that was rotating about its center at a constant angular speed, with a moving (infinitesimal) ball inside. The ball had a constant linear speed, and there was no friction or gravity. When the ball hit an edge of the square, it simply reflected as though the square was momentarily stationary during the briefest of moments they were in contact. Also, the ball was not allowed to hit a corner of the square—it would get jammed in that corner, a situation we preferred to avoid.

Suppose the ball was traveling on a periodic (i.e., repeating) path, and that it only ever made contact with a single point on the unit square. What was the shortest distance the ball could have traveled in one loop of this path?

Whenever the ball hit a wall, you could break its motion down into two components—a “normal” component, which represented the ball’s perpendicular motion directly into and away from the wall, and a “transverse” component, which represented the ball’s motion parallel to the wall.

To minimize the motion of the ball, it made sense that the transverse component was zero. In other words, the ball bounced directly into the wall each moment it made contact. As a result, it bounced back and forth between opposite ends of the circle traced out by the midpoint of one of the square’s sides, as shown in the following video:

One complete cycle for the ball was down a segment that has the same length as a side of the square, and then back up again. Since this was a unit square with side length 1, the total path length was 2.

Several solvers noted that the square had to have just the right angular speed for the ball to keep hitting the same midpoint over and over again. Suppose the ball moved at a speed of 1 unit per second. That meant it took 1 second for the midpoint to make one half-revolution around its circular path, so the angular speed was 0.5 revolutions per second, or 0.5 Hz.

Higher angular speeds also worked. Instead of making 1/2 revolutions each second, the square also could have made 3/2 revolutions, 5/2 revolutions, 7/2 revolutions, etc. Here’s what things would have looked like with a 1.5 Hz rotating square:

Last Week’s Extra Credit

Congratulations to the (randomly selected) winner from last week: 🎻 Rohan Lewis 🎻 from Cary, North Carolina. I received 36 timely submissions, of which 28 were correct—good for a 78 percent solve rate.

Again, you had a rotating unit square and bouncing (infinitesimal) ball inside.

As we just found, the shortest repeating path for which the ball made contact with a single point on the square had length L₁ = 2.

The next shortest repeating path for which the ball made contact with a single point on the square had length L₂. To be clear, L₂ > L₁.

What was the length L₂?