“Guess How Many Times Your Heart Will Beat” argued that careful guessing can be taught. Every school fair offers a chance to prove it: the jar of sweets on the table at the front, a prize for the closest guess. Most people stare at it for five seconds and write down a number that feels right. You can do much better, with one sum and one number worth remembering.
Start the way the speech said, by breaking the problem into pieces you can estimate. The jar first. Say it’s roughly a cylinder, 12 centimetres across and 20 centimetres tall. The volume of a cylinder is π times the radius squared times the height, so that’s about 3.14 × 6 × 6 × 20, which comes to around 2,260 cubic centimetres. Then one sweet: a round one about 2 centimetres across has a volume of about 4 cubic centimetres. Divide one by the other and you get about 560 sweets.
That’s the number most careful people write down, and it’s wrong, by a lot. Because sweets aren’t liquid. They can’t fill every corner. Between round objects there are always gaps, and a surprising amount of a jar of sweets is air.
How much? This is a famous problem in mathematics, and it’s harder than it looks. In 1611, the astronomer Johannes Kepler asked how tightly you can possibly pack identical balls, and guessed that the answer was the arrangement greengrocers use for oranges: neat layers, each ball nestled in a hollow of the layer below. That fills about 74 per cent of the space. It seems obvious that nothing could beat it, but proving it took nearly four hundred years, and the final proof, completed in the 2010s, needed computers to check thousands of separate cases.
But your sweets aren’t stacked neatly. They were poured in. Randomly poured balls can’t find those perfect hollows; they jam against each other at awkward angles and leave bigger gaps. Shake them down as tightly as random packing allows and they fill about 64 per cent of the space. Pour them in loosely without shaking and it’s nearer 60. So take your 560 and multiply by about 0.64. You get roughly 360. That single step, which almost nobody at the fair takes, moves your guess by two hundred sweets.
Shape matters too, in an unexpected way. You might think flattened sweets, like the coated chocolate ones shaped like little discs, would pack worse than perfect balls. In 2004, physicists poured huge numbers of them into containers and measured: they actually pack better, about 68 per cent, because their slight flatness lets them tilt and fill gaps that a sphere can’t. So for that kind of sweet, use 0.68.
The jar itself has one more effect. Next to the flat glass wall, sweets can’t nestle into each other the way they do in the middle, so there’s a layer of extra-large gaps all round the edge. In a big jar that hardly matters. In a small jar packed with big sweets, a large share of the sweets are touching the glass, and the packing fraction drops. So if the sweets are large compared with the jar, shade your guess down a little.
The speech mentioned one more idea worth using here: that errors partly cancel. In 1907, the scientist Francis Galton went to a country fair where about eight hundred people were guessing the weight of an ox. Individual guesses were all over the place. But when he lined them up in order, the middle guess was within one per cent of the real weight. Many independent guesses, each wrong in its own direction, cancel each other out. So if you’re allowed, quietly ask a few friends for their guesses, take the middle one, and then compare it with your calculation. If the two are close, you can be confident. If they’re far apart, check your measurements of the jar.
Then write your number down and wait. You might not win: someone might get lucky. But you’ll have done what the speech asked for. You’ll have turned a feeling into a method, and you’ll know exactly why your answer is the size it is.