Here's the hardest "scale puzzle" in my collection:

You have twelve identical coins. They all look the same, but one is a counterfeit. It is either heavier or lighter than any of the others. Use a balance scale three times to figure out which coin is counterfeit.

• Scale Puzzles 101: You have three coins, fake is heavier, use scale once to find fake.
• Scale Puzzles 102: You have two coins, one is fake, but you don't know if fake is heavier or lighter. You also have one genuine coin. Use scale once to find fake.
• Scale Puzzles 201: You have nine coins, the fake is heavier. Use scale twice to find fake.

• Divide coins into three groups of four coins
• Use the method from SP101 and SP201 to compare the first group to the second group.
• This will tell you one of two things, either:
  • If scale balances, the fake's in the 3rd group (so figure out a method to find the fake out of those four coins in two more weighings)
    • For your second weighing, you're looking to figure out if the fake is in a particular subgroup of two or three coins.
    • For your third weighing, you'll use the method from SP101 or SP102.
  • If the scale isn't balanced, the 3rd group is genuine, so figure out a weigh to find it from the first eight coins in two more weighings.
    • For your second weighing, figure out a method that lets you evaluate three groups of three at once, while weighing four coins at a time.
    • For your third weighing, you'll use the method from SP101.
  • Remember, the first weighing tells you that some coins are genuine, so use that information as part of your remaining strategy.

This entire solution method is complicated to explain, so let's look at every step, and every possible outcome. In this detailed step-by-step, I will use the letters (A, B, C, D, E, F, G, H, J, K, L & M) to represent the coins. Let's get started:

• Divide the coins into three groups
  A+B+C+D, E+F+G+H, and J+K+L+M
• Put A+B+C+D on the left side of the scale
• Put E+F+G+H on the right side of the scale
• See if they balance

Figure out what each possible outcome will tell you about the coins on the scale, and about the coins not on the scale, before you click the link for that outcome to see my explanation.

Did the Scale balance? See choices to right --->

The scale balanced

_(A+B+C+D)_ /\ _(E+F+G+H)_

The right side went down

^_(A+B+C+D)_ /\ _{_(E+F+G+H)_}