USPuzzle Contest: 1 – Measuring Puzzles

This is a set of puzzles revolving around clever measuring tricks.  Each of the three puzzles is worth one point.

1. You have a balance scale with 2 pans.  If you have 2 identical-looking spheres of different weights, you can find which is heavier with 1 use of the scale (one sphere on each side).  If you have 4 identical-looking spheres, 1 of which is heavier than the others, 2 uses of the scale are necessary to guarantee identifying the heavy one.

Now: You have 9 identical-looking spheres, 1 of which is heavier than the rest (Edit: The other 8 spheres all weigh the same.).  What is the fewest uses of the scale necessary to guarantee the identity of the heavy sphere, and how do you do it?

2. You have a single scale that measures accurately in grams.  You also have 11 stacks of 10 coins each.  One of these stacks is made entirely of counterfeit coins made with a denser base metal (say lead) such that a counterfeit coin weighs 1 gram more than a genuine coin.  If you, as an agent of Standards and Measures, already know by heart the correct weight of a real coin: What is the fewest times you must use the scale to guarantee you know which stack of coins is counterfeit?  And how do you do it?

3.  You have a balance scale with 2 pans.  You also have 6 spheres: 2 red, 2 yellow, and 2 blue.  1 sphere of each color is heavy, and 1 sphere of each color is light.  All the heavy spheres weigh the same as each other, and all the light spheres weigh the same as each other.  What is the fewest times you must use the scale to guarantee you identify all the heavy spheres?  And, of course, how do you do it?

Submit solutions to usp.blogmatters@gmail.com, see rules post below for details.