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First we need to find some commonalities between the gumball and the jar. Both the gumball and the jar can be converted into cubic inches.

The formula for finding the volume of a sphere is V = 4/3piR³

Assuming the 1" dimension is the diameter of the gumball.

Then:

V = 4/3pi(½)³

v = 4/3 x 3.14 x (1/2)^3

v = 4/3 x 3.14 x .125 = 0.52 cubic inches

Also assuming the interior of the jar is one gallon then using a conversion chart we can find that one gallon is equal to 231 cubic inches.

We could divide 231 by .52 to get roughly 444, however this does not account for the empty space between each gumball. For estimating, we will assume that each 1 inch gumball with the surrounding empty space is equal to 1 cubic inch. Therefore, in a 1 gallon jar which will hold 231 cubic inches will hold 231 gumballs.

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Well... that answer is very intuitively nice, but the empty space approximation is very messy, and gives each gumball a lot more room than it needs.

The best possible packing of spheres fills about 74% of space, so we can say that there are at most about 328 gumballs in the jar. However, unlike circles, spheres don't automatically fall into a regular close packing when allowed to settle; three dimensions allow them to "get in each other's way" in a manner that would require them to move apart before they can move closer together. Experiment has shown that a random dense packing has about 64% density. So a better estimate might be (64% of 231)/.52 = 284 gumballs.

Given the way people actually pack gumballs into jars--add them, shake the jar, add more if they fit now--I would go with a number between 284 and 328, if I were entering a drawing.

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You've got two great mathematical answers below, but what about using a little simple statistics?

If you get lots (and by lots you probably need more than 100) of guesses, add them all together and divide by the total number of guesses to get the average (or mean) of the guesses it may well turn out to be surprisingly close to the actual answer.

This idea of the wisdom of the crowds was first 'discovered' by Francis Galton in 1906 when a crowd had to guess the weight of a cow - the average guess was accurate to within 0.1% of the actual weight of the cow.

I'm trying to recreate the experiment online of guessing the cow's weight - I've written a hub about it and a interactive website on which you can join in the experiments.

One test is to guess the amount of smarties (small choclate sweets, like M&Ms) in a pint glass - why not have a go?!

Thanks for reading,

Jeff

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