Skip to Content
Artificial intelligence

The Curious Mathematics of Domino Chain Reactions

A toppling domino can push over a larger domino but how much bigger can the next one be? One mathematician thinks he’s worked out the secret behind domino chain reactions

You’ve probably seen the domino effect in action where a row of standing slabs topple successively. Usually, the dominos are all the same size but a toppling domino actually has enough momentum to push over a bigger one. So it’s possible to set up a row of successively larger dominos that can be toppled by the push of a tiny slab at the beginning–a domino chain reaction.

So here’s an interesting question. How much bigger can each succeeding domino be?

Today, J M J van Leeuwen at Leiden University in The Netherlands takes this problem by the scruff of the neck and gives it a good mathematical shaking. It turns out that the answer–the maximum growth factor– isn’t quite as simple as the problem might suggest.

There are various videos, such as this one, on the internet that give a good demonstration of the chain reaction effect. The standard thinking is that a domino can topple another about 1.5 times as big, provided that the spacing between them is optimal.

The basic physics is straightforward. Standing a domino on its end stores a certain amount of potential energy which is released by pushing it over. However, the force required to topple the domino is smaller than the force it generates when it falls. It is this “force amplification” that can be used to topple bigger dominoes.

But the devil is in the detail since there are various ways that the dominoes lose energy as they topple. For example, a toppling domino comes to rest on its neighbour. So the collisions are inelastic, which is the main source of lost energy.  And in practice, the dominos can slide along the floor when they are hit and this can seriously impede toppling.

So van Leeuwen makes a series of simplifications in his mathematical analysis. He assumes that the friction between the ground and the dominoes is effectively infinite so that they cannot slide. He assumes the collisions are fully inelastic so the dominoes stay in contact with each other when they collide. He also assumes that once in contact with each other, the dominoes slide frictionlessly over each other. 

Given these assumptions, he then shows that with optimal spacing, each succeeding domino can be no more than about twice as big as the previous one, which is maximum growth factor of no more than about 2. 

That’s significantly more than has been assumed in the past. He admits that achieving this limit is probably unrealistic in practice because the assumptions can never hold perfectly. For example, dominoes will always slip by a small amount. 

Nevertheless, even a growth factor of 1.5 leads to some extraordinary chain reactions. A series of 13 dominoes that grow at this rate will amplify the force needed to push the smallest by a factor of 2 billion. And it doesn’t need a particularly long series before the largest dominoes are the size of sky scrapers.

Entertaining mathematics!

Ref: arxiv.org/abs/1301.0615: Domino Magnification

Deep Dive

Artificial intelligence

The inside story of how ChatGPT was built from the people who made it

Exclusive conversations that take us behind the scenes of a cultural phenomenon.

ChatGPT is about to revolutionize the economy. We need to decide what that looks like.

New large language models will transform many jobs. Whether they will lead to widespread prosperity or not is up to us.

AI is dreaming up drugs that no one has ever seen. Now we’ve got to see if they work.

AI automation throughout the drug development pipeline is opening up the possibility of faster, cheaper pharmaceuticals.

GPT-4 is bigger and better than ChatGPT—but OpenAI won’t say why

We got a first look at the much-anticipated big new language model from OpenAI. But this time how it works is even more deeply under wraps.

Stay connected

Illustration by Rose Wong

Get the latest updates from
MIT Technology Review

Discover special offers, top stories, upcoming events, and more.

Thank you for submitting your email!

Explore more newsletters

It looks like something went wrong.

We’re having trouble saving your preferences. Try refreshing this page and updating them one more time. If you continue to get this message, reach out to us at customer-service@technologyreview.com with a list of newsletters you’d like to receive.