Mathematical Analysis Lends Insight to Longstanding Magic Trick
By Lina Sorg
In addition to delighting and confounding people of all ages, magic tricks can be used to demonstrate principles pertaining to mathematics or physics. The handson nature of these tricks make them valuable visual resources for the explanation of otherwise abstract concepts. During a minisymposium presentation at the 2019 SIAM Conference on Applications of Dynamical Systems, currently taking place in Snowbird, Utah, Jared Bronski of the University of Illinois offered a mathematical explanation of a longstanding magician’s trick involving a ball chain and steel ring.
Bronski opened his presentation with a demonstration of the magic trick, which he learned 20 years ago as a postdoctoral researcher. “I used to use it to win bets in bars but now I use it to impress my calculus students, which is a brutal commentary about the march of time,” he quipped. The trick requires two unassuming components: a ubiquitous metal ball chain and a steel ring. Bronski threaded the ring onto the loop of chain, then let it go. Instead of clattering to the floor as one might expect, the ring ended up caught in a knot at the bottom of the chain.
Jared Bronski adjusts his tabletop ring dropper to demonstrate the mathematical concepts of the classic magic trick involving a ball chain and steel ring.
How does this happen? Bronski assured the audience that the trick is highly reproducible once the “magician” knows the right initial conditions. He began by pointing out a few general observations about the circumstances surrounding the trick, including the fact that the chain and ring are obviously not topologically linked. He also noted that parameters seem fairly insignificant (with a few exceptions), as the trick works well with a variety of ring sizes and chain lengths/thicknesses. This suggests that the socalled magic actually has something to do with physics.
Bronski next presented the audience with a number of variables that might hypothetically matter in this type of situation, including the ring’s mass, the length and radius of the chain, and the density of both the surrounding air and the steel ring. While most of these factors seem to be negligible, the masses of the chain and ring are significant. The trick does not work if the chain has a very small mass; for instance, Bronski has tried and failed to successfully complete it with a string (of much lighter mass) standing in for the chain. The relationship between the mass of the ring and the chain is also significant. “It tends to work best if the mass of the chain is smaller than the mass of the ring,” he said, adding that the ring’s mass must be considerately greater than that of the chain (which itself cannot be insignificant). This need for substantial mass suggests that an inertial process related to wave propagation is likely occurring.
Bronski then suggested the following fivepart mechanism to explain the tricks’ nuances:
 The ring is dropped
 The ring interacts with the chain and excites a wave
 The ring falls while the wave propagates down the chain
 The ring catches up with the wave it previously excited
 The ring passes through the wave, completing the “catch.”
Knowing this information makes the trick even more reliable, and Bronski passed around rings and chains so audience members could try it themselves. “What you really need is asymmetry,” he said amidst the clinks of rings hitting the floor in failed attempts. He elaborated on this point, explaining that the ring will not excite a wave if it is dropped perfectly down the chain. Alternatively, Bronski suggested focusing on the release. After threading the ring on the chain and roughly centering it, he pushed it away with his index finger, causing it to momentarily stick to his thumb as it tumbles down. This slight bit of asymmetry propagates the wave, makes the ring want to rotate on its way down, and ultimately causes the end of the chain to flip up and knot itself on the ring.
To further validate these movements, Bronski employed a tabletop ring dropper that he crafted himself. After reaffirming that the wave equation works well with a ball chain but not with string, he reminded attendees that the ring accelerates as it falls due to gravity. Because the chain hangs under gravity, tension is not constant and instead gets smaller towards the bottom of the chain because less mass is pulling on it.
Bronski concluded his presentation with one final inperson demonstration and highspeed video footage that supported his interpretations. Once again, the ring caught up to and interacted with the decelerating wave that it previously excited. “The ring always has time to overtake the disturbance,” he said. “Magic is all about belief.”

Lina Sorg is the associate editor of SIAM News. 