To estimate how many jelly beans are in a jar, work out the jar's volume, multiply by about 0.65 because beans only fill around two thirds of the space, then divide by the volume of one bean (roughly 1 millilitre for a regular jelly bean).
The guessing jar at the school fair is a maths problem in disguise. Here is how to win it.
Most people guess the jelly bean jar by staring at it and picking a number that feels right. That usually goes badly. People tend to undercount a full jar, because they judge the front face they can see and forget how deep it goes.
You can do much better with three numbers and a phone calculator. The method works for jelly beans, sweets, marbles, coins or anything else piled into a container.
Count = jar volume x packing fraction / volume of one bean
Most guessing jars are cylinders. Measure the inside width and the height of the beans, not the height of the lid.
Volume of a cylinder = 3.14 x radius x radius x height
A jar 10 cm across and filled 15 cm high has a radius of 5 cm, so its volume is 3.14 x 5 x 5 x 15 = about 1,180 cubic centimetres, or 1,180 millilitres.
If you can't touch the jar, use a known object for scale. A standard credit card is 8.6 cm long. Hold one up at arm's length next to the jar, or compare against a hand you have already measured. Our guide to measuring without a ruler lists ten other objects with known sizes.
If the jar is printed with its capacity (a 1 litre or 32 oz jar, say), use that and take off any empty space at the top.
Beans do not fill a jar the way water does. Round and oval things poured in at random fill a bit under two thirds of the space. The rest is air.
| What is in the jar | Rough share of space filled |
|---|---|
| Marbles or round sweets | 0.60 to 0.64 |
| Jelly beans, chocolate buttons | 0.64 to 0.70 |
| Coins, loose | 0.60 to 0.65 |
| Shaken and settled well | add about 0.03 |
A value of 0.65 is a safe default for jelly beans. In our example jar that leaves 1,180 x 0.65 = about 770 millilitres of actual beans.
A regular jelly bean is about 2 cm long and 1.5 cm wide, and takes up roughly 1 millilitre. Small gourmet beans are a little smaller. Big supermarket ones can be closer to 1.5 millilitres.
If you can get a matching bean, the best trick is water. Pour water into a measuring jug, drop in 20 beans, and read how far the level rises. Divide by 20. You now know your bean to within a few percent.
Back to the example: 770 / 1 = about 770 beans. With bigger 1.3 ml beans it would be 770 / 1.3 = about 590.
Before you write your guess, count a single layer. Count the beans you can see along the bottom row of the glass, then count how many rows go up the side.
Two methods that land near each other are a good sign. If they are miles apart, recheck the jar size first. That is almost always the mistake.
There is a famous version of this from 1906. At a country fair in Plymouth, England, around 800 people guessed the weight of an ox. Francis Galton collected the tickets and found the middle guess was 1,207 pounds. The ox weighed 1,198. Individuals were way off. The crowd was within one percent.
If you are guessing with friends, collect everyone's number and take the middle one. It usually beats almost every single guess, including yours.
Estimation is a skill, and it gets better with feedback. The problem with jars is you only find out the answer once a year.
Halfsies gives you that feedback every day. You get three hand-drawn objects and one cut each, and you try to slice each one exactly in half by eye. A scale weighs both halves, and your score is the lighter half divided by the heavier half, times 100. It takes about two minutes and it is free in your browser, on iPhone and on Android.
Multiply the jar's volume in millilitres by 0.65 and divide by about 1, since a regular jelly bean takes up roughly 1 millilitre. A 1 litre jar holds somewhere around 650 regular jelly beans.
A US quart is about 946 millilitres. At 0.65 packing and 1 millilitre per bean, that is roughly 600 regular jelly beans, or about 450 if the beans are large.
They judge the beans they can see on the outside of the glass. The number of beans grows with the volume, which rises much faster than the visible surface as a jar gets bigger.
Yes, a little. Shaking lets beans settle into gaps, so more fit in the same space. Expect about 3 to 5 percent more beans in a jar that has been tapped down.