Counting Matchsticks by Listening

How accurately can you determine the number of matchsticks in a matchbox from the sound it makes when you shake it ? Can the same methods be applied to a matchbox containing chewing gums ?

Such read one of the questions from the International Physics Tournament (IPT), 2023. This was the problem that I, along with my friends tried to solve as part of our college team that year. In this blog post, I document our journey trying to solve this problem.

On the face of it, there seems to be very little information and very many ways to solve this problem. We started with the notion that our solution should apply to matchsticks and chewing gums alike, hence we couldn’t make any geometrical assumptions. We tried applying all the theory we had learnt in 4 semesters of Physics so far, from Mean Free Path to Fourier Analysis… to no avail.

As theory wasn’t working, we tried an experimental approach. Could we find an empirical formula ? We worked our intuitive muscles hard and arrived at the following result :

Before I explain this formula, we need to remember a few basic facts about sound : (i) sound is a pressure wave (ii) sound needs a medium to travel. In our case sound is produced when the walls of the matchbox vibrate, thereby creating a pressure difference which is heard outside. This happens in two ways : (i) matchsticks collide directly with the walls (ii) two matchsticks collide with each other. The other thing that matters is the amount of air left inside the box.

Hence the total sound produced ‘S’ depends on the sum total of pressure difference created by each matchstick and the volume of air left inside the box. The total pressure difference created is assumed to be linearly related to the number of matchsticks present. The edge cases agree with this formula. If the matchbox is empty and you shake it, you hear nothing (the first term is 0). If the matchbox is completely filled and you shake it, you hear nothing (the second term is 0).

This implied that maximum sound is produced with an optimal balance of number of matchsticks and volume of air left in the box. So if we plot the sound heard as a function of the number of matchsticks, we would see a rise at the start, a peak and drop at the end. For no good reason, we assumed a Gaussian distribution. Physics being an experimental science, we set up a means of testing our hypothesis.

We wanted to see how different types of material (plastic, wood, tic-tac) affected our results, also the size of the matchsticks and matchbox itself. We had to glue our matchbox to a scale fixed vertically, pull the scale back a set distance and record the sound produced on a high quality mic for fixed time intervals. All that was left to do was repeat this by varying the number of matchsticks & analyzing the data ! Given below are some of our better results.

Medium sized matchbox filled with small plastic matchsticks

Small matchbox filed with small plastic matchsticks

So far so good. Experiments seemed to agree with our assumptions. But perhaps the most important outcome of our study was establishing that the type of matchstick doesn’t affect the sound produced ! The graphs showcasing that are attached below. In the first one we see a discrepancy for the number of matchsticks due to their varying sizes (bigger matchsticks fill faster). When adjusted for volume, we see that when 40% of the matchbox is filled, it leads to maximum sound, irrespective of what you fill it with.

Different matchsticks displayed by number

Different matchsticks displayed by volume

While we couldn’t answer the question entirely or progress in the IPT Tournament that year, this project gave us a chance to solve a problem by ourselves, one that did not have back of the book solution. It was our first experience in ‘Physics research’. I am grateful for getting a chance to work on solving this problem, both to my college and my team mates Satya and Ninad.

The final, comprehensive document that we submitted can be found here. If you want to have a conversation about the problem or suggest improvements / alternate solutions, do reach out !