Some problems have a solution that is easy to find, but the best solution is hard to find. The purpose for this homework is to look for the best solutions.


1) There are 10 gumball machines. Most of the machines have gumballs that weigh 1 gram, but one of the machines has gumballs that weight 1.2 grams. You can't tell the difference between them just by holding them. You need to use a scale.

You can take as many gumballs out of the machines as you need. How can you determine which machine has the heavier gumballs while using the scale as few times as possible?







2) This time you have a balance instead of a scale. The way a balance works is that you place items in each side, and the balance will tilt toward whichever side is heavier. You have 8 marbles. Most of them have the same weight, but one marble is slightly heavier than the others. Again, you can't tell the difference just by holding them.

There is a way you can tell which is the heavy marble while using the balance only twice. How?







3) Now you have 12 marbles. Most of them have the same weight, but one marble is slightly different. You don't know if the different marble is heavier or lighter than the others. How can you tell which is the different marble while using the balance only three times?