How to estimate the number of 1-inch gumballs in a 1 gallon jar

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.

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.