How Big a Domino Can a Small Domino Knock Over?

The first dominoes

The first domino is 8.1 mm tall. Each domino is 1.5 times as tall as the one before.

Look along your table. The 2nd domino is 12.2 mm and the 8th is 138.9 mm. The 3rd is 18.3 mm and the 9th is 208.3 mm.

That rule of thumb lets you predict without a calculator. A domino 10 times the first is about 6 dominoes on, 100 times about 12 on, 1,000 times about 18 on (1.5¹⁸ is 1,478; 1.5¹⁷ is 985).

A geometric sequence

The heights 8.1, 12.2, 18.3, 27.4… are a geometric sequence: each number is the one before multiplied by the same number, here 1.5. That number is the common ratio, r. You can describe the chain in two ways.

• Step by step: the first domino is 8.1 mm, and each next one is 1.5 × the one before. That is how you build the chain, and how you filled in the table.
• In one go: domino n has been multiplied by 1.5 exactly n − 1 times (domino 1 not at all, domino 2 once), so with Grade 8’s indices

$$\text{height of domino } n = 8.1 \times 1.5^{\,n-1} \text{ mm.}$$

The second way jumps straight to any domino without the ones in between: domino 19 is 8.1 × 1.5¹⁸ = 8.1 × 1,478 ≈ 12,000 mm, 12 m. The Grade 10 book writes any geometric sequence the same way, $t_n = ar^{n-1}$, with a the first term Grade 10: 6.0 Review.

Growth like this, multiplying by the same number at every step, is exponential growth, the growth of the rice on the chessboard in topic 1 (where r was 2). Here r is smaller, so the chain grows more slowly than the rice, but it still runs away in the end.

Adding would take forever

Suppose instead that each domino were only 8.1 mm taller than the one before: 8.1, 16.3, 24.4, 32.5 mm… That is a sequence where the same number is added each time, an arithmetic sequence, growing in a straight line (linear growth). It sounds about as fast at first: domino 2 is 16 mm instead of 12. But to reach 10 m it needs 1,231 dominoes. Multiplying by 1.5 needs 19.

It isn’t that 1.5 is a big number. It is that multiplying lets each step grow with the domino: the step from domino 18 to 19 is 4 m, while the first step was 4 mm. An adding chain takes the same small step forever.

Here is the surprise of the exhibit. A tiny push on the first domino ends with a bang that shakes the floor. Where does all that energy come from?

Energy grows as size × size × size × size

You met potential energy in Grade 6 Science Grade 6: Forms of energy: something lifted up stores energy, and the energy is its weight times the height it can fall. For a domino, the height that matters is the height of its center, the middle of its wood, which drops when it falls.

Now make a domino twice as big in every direction.

So doubling the size multiplies the energy by 2 × 2 × 2 × 2 = 2⁴ = 16. In the same way, making a domino 1.5 times bigger multiplies its wood by 1.5 × 1.5 × 1.5 = 3.375 and the height of its center by 1.5, so its energy by

$$1.5^4 = 5.0625,$$

about 5 times each step. The heights in the chain go up 1.5 times a step, the energies 5 times: another geometric sequence, with a common ratio of 1.5⁴.

Where the energy comes from

The energy isn’t in your push. It was put into each domino when somebody stood it up, lifting its center from lying down to standing. Your push only lets it out. To see why a small push is enough, tip one over.

When a domino tips, it turns about its front bottom edge. Its center swings round that edge in a circle and first goes up, a tiny bit, until it is straight above the edge: that is the tipping point. Let go before it and the domino rocks back. Past it, the center goes down, and the domino falls by itself.

How high is the center at the tipping point? It is half the domino’s diagonal above the floor. With Pythagoras’ theorem (topic 18), for a domino of height h and thickness t that is

$$\tfrac{1}{2}\sqrt{h^2 + t^2},$$

only a little more than $\tfrac{1}{2}h$, where it started. For a domino whose thickness is ⅛ of its height, the center rises just 0.39% of the domino’s height to reach the tipping point, and then drops 44% of it to the floor, 113 times as far. For the 8.1 mm domino, your push lifts its center by about 0.03 mm.

So a domino is a trigger: a little energy in, a lot out. Each falling domino has far more energy than it needs to give the next one its little push, even when the next one is bigger. That is why a chain of growing dominoes can keep going, and why its energy can keep growing.

To the top of the world

In 2012 the Dutch Science Quiz asked its own version: how many growing dominoes to topple one as tall as the Domtoren, the cathedral tower of Utrecht, 112 m tall? From our 8.1 mm domino at 1.5 a step, the 24th domino is 91 m and the 25th 137 m. (The quiz started from an ordinary domino, which is taller than ours, so it needs fewer.)

Is the last domino taller than all the others?

On the chessboard, each square held one grain more than all the squares before it put together. Is the last domino taller than all the dominoes before it, stacked one on top of another?

Here is why, and how far it goes. Walk backwards from the last domino. The one before it is $\tfrac{1}{1.5} = \tfrac{2}{3}$ of its height, the one before that $\tfrac{2}{3}$ of that, $\tfrac{4}{9}$, and so on:

$$\tfrac{2}{3} + \tfrac{4}{9} + \tfrac{8}{27} + \tfrac{16}{81} + \cdots$$

This sum never quite reaches 2, however many dominoes there are, just as $\tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \cdots$ never quite reaches 1 (topic 33). For the Toronto chain of 13 it is 1.98: the 12 smaller dominoes, stacked, are nearly twice as tall as the 13th. The sum of a geometric series says the same in one line: the first n terms of $a, ar, ar^2, \ldots$ add up to

$$a \times \frac{r^n - 1}{r - 1}.$$

Why? Multiply the whole sum by r. Every term moves up one place, so r times the sum has all the same terms except that it gains $ar^n$ at the top and loses a at the bottom. The difference, r times the sum take away the sum, is (r − 1) times the sum, and it is $ar^n - a$. Divide by r − 1.

So the rule of the chessboard works for chains of every length only when r is 2 or more. With r = 2 the earlier squares add up to one less than the next one; with r = 1.5 they add up to nearly twice it.

The twist: nothing grows for ever

Nobody could make a domino as tall as Everest, or stand it up. And the chain can’t go on getting smaller backwards either: 12 dominoes before our 8.1 mm one, the domino would be 0.06 mm tall, thinner than a sheet of paper (about 0.1 mm). Exponential growth is a good description of the chain on the floor at Science Playground, and nonsense far outside it. That is true of almost every exponential growth in the world (topic 1’s bacteria too): always ask what runs out.

The growing domino chain is usually traced back to a one-page note by the physicist Lorne Whitehead in the American Journal of Physics in 1983, which described one with each domino 1.5 times the one before. Chains like it are now shown in physics classes (the University of Toronto’s has 13 dominoes), at science centers (the Exploratorium in San Francisco built one for an evening of chain reactions in 2001), and at ours.

See the chain for yourself at Science Playground: Growing Dominoes, among the maths exhibits.

• How Much Rice Is on the Chessboard? Doubling instead of 1.5, and why each square holds more than all the ones before.
• Can You Add Halves Forever and Still Get One? Sums like ⅔ + 4⁄9 + 8⁄27 + … that never reach their end.
• Why Is a Magnitude 8 Earthquake Not Just a Bit Bigger Than a 7? (Grades 9–12) Another scale where energy grows by multiplying: about 32 times a step.
• What Is e, and Why Does Growth Keep Finding It? (Grades 9–12) Growth that multiplies every moment, not in steps.
• How Many Moves to Move 64 Disks? (Computing, Going Further) Another doubling, and you program it.
• Grade 10, 6.4 Sum of a geometric series: the formula for adding up any chain like this one.