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How to guess the number of sweets in a jar
Guess the sweets in the jar is a familiar game at school fairs and fêtes. You can do much better than a wild guess. Here is a simple way to estimate the number, what scientists learned about packing M&M's, and why the middle of everyone's guesses can be surprisingly close.
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Guessing the number of sweets in a jar feels like luck, but a little maths gets you much closer. The idea is simple: work out how much space is inside the jar, allow for the gaps between the sweets, and divide by the space one sweet takes up.
Step 1: the space inside the jar
For a straight sided, round jar, measure the inside width and the height the sweets reach. Halve the width to get the radius, then work out π × radius × radius × height. With measurements in centimetres, that gives cubic centimetres, and 1,000 cubic centimetres is a litre.
Step 2: allow for the gaps
Sweets never fill a jar completely, because there are gaps between them. Experiments and computer simulations show that same size spheres, packed at random, fill about 64% of the space at most. So for roughly round sweets, multiply the jar's volume by about 0.64.
Shape matters. In 2004, researchers at Princeton and other universities found that ellipsoids, squashed spheres with a shape close to that of M&M's, can pack more densely, filling about 68 to 71% of the space. One of the team, Salvatore Torquato, pointed out that M&M's made good test objects because you could eat the experiment afterwards. A jar of mixed sizes can pack differently again, because small pieces fill the gaps between big ones.
Step 3: divide by one sweet
Estimate the volume of one sweet. For a round sweet, work out 4 ÷ 3 × π × radius × radius × radius. Divide the filled space from step 2 by that, and you have your estimate.
An example. Say a jar holds 1,000 cubic centimetres of round sweets, and each sweet is about 2 cubic centimetres. The sweets fill about 1,000 × 0.64 = 640 cubic centimetres, and 640 ÷ 2 = 320 sweets.
A cross check: count the layers
Count the sweets you can see along the bottom of the jar and up the side, and use them to estimate how many fit in each layer and how many layers there are. If this and the maths roughly agree, you can be more confident.
Weighing is easier, if you can
In a shop, sweets are sold by weight, and weight makes counting easy: divide the total weight by the weight of one sweet. Barratt's Black Jack chews, for example, come 46 to a 250 g pack, which works out at about 5.4 g a chew: see how many sweets are in 100 g of pick and mix?
The wisdom of the crowd
If you can, ask other people too, and take the middle guess. At a country fair in Plymouth in 1906, around 800 people guessed the weight of an ox. The statistician Francis Galton found that the middle guess, 1,207 pounds, was within 1% of the true weight of 1,198 pounds. Crowds do best when there is a right answer and people guess independently, and a jar of sweets has a right answer.
Citable formats
For journalists, researchers, AI assistants and content creators. Pick the format you need:
Free to cite, quote, and reuse with attribution to sweeties.net.
Questions and answers
How do you estimate how many sweets are in a jar?
Work out the volume inside the jar, multiply by the fraction of space the sweets fill, about 0.64 for roughly round sweets, then divide by the volume of one sweet. Counting the sweets along the bottom and up the side is a good cross check.
How much of a jar do sweets actually fill?
Never all of it, because of the gaps. Randomly packed spheres fill about 64% of the space. A 2004 study found that shapes like M&M's can pack more densely, filling about 68 to 71%.
Is it better to guess on your own or with others?
If several people guess independently, the middle guess can be very close. At a country fair in 1906, the middle guess of 800 people for the weight of an ox was within 1% of the true weight, an example of what is called the wisdom of the crowd.
Sources
- Random close pack, Wikipedia
- Improving the density of jammed disordered packings using ellipsoids, Donev and others, Science, 2004, Science
- Candy science: M&Ms pack more tightly than spheres, 11 February 2004, Science News
- Wisdom of the crowd, Wikipedia
- Black Jack (confectionery), Wikipedia
Keep reading
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