Coin Balancing Problems

Nine Coins - difficulty: medium
The Setup: You have nine coins that are identical in weight save for one, which is lighter than the others—a counterfeit. The difference is only perceptible by using a balance, but only the coins themselves can be weighed, and it can only be used twice in total.

The Problem: Is it possible to isolate the counterfeit coin with only two weighings?

The Solution: Divide the nine coins into three groups of three. Weigh one group against another. If the first two groups balance then the counterfeit coin is not there. Then you can take the third group and put one coin on each side. If it balances, you hold the counterfeit coin. If it doesn't balance, then the lighter coin is the counterfeit one. If the original groups of three do not balance, then put two coins from the lighter group on the balance and whichever side is lighter holds the counterfeit (or the third coin is the counterfeit if the two balance, by process of elimination.)

Twelve Coins - difficulty: hard
The Setup: You have twelve coins and a balance scale. One of the coins does not weigh the same as the other eleven, but you don't know if the odd coin is heavier or lighter than the others. The difference is only perceptible by using a balance, but only the coins themselves can be weighed, and it can only be used three times in total.

The Problem: How can you determine, in three weighings, which coin is the counterfeit?

The Solution: First weigh four coins against four coins. If they balance, you know the counterfeit is among the remaining four and can deduce it in two weighings by comparing three of the suspect coins with three you know to be legitimate (from the first weighing); this will tell you whether the counterfeit is heavier or lighter and then you can repeat the final step from the nine coin problem to isolate it.

Alternatively, if the first weighing produces uneven sides the solution is a little trickier. We now have eight suspects instead of merely four and will have to investigate more carefully. We essentially have two different tactics at our disposal:

Tactic One - Weigh the Lighter Group against a Control Group
The trick is to use the information we already know: that one group is lighter than the other group. If we swap the heavy group out for the coins we know to be legitimate (the unweighted group from the first weighing) and "the lighter group" is still lighter, we know it holds the counterfeit and that the counterfeit is lighter. Alternatively if "the lighter group" and the legitimate coins balance, the heavy group must have held the counterfeit (which would have to be heavier). But this method will only narrow the candidates down to four coins...still one too many to solve the problem with the single weighing we have left, so this tactic alone is insufficient.

Tactic Two- Swap coins from Group X and Group Y
If we swapped one coin from the heavier group and one from the lighter with each other, either the scales will tip the other way or stay the same (they can't balance because the counterfeit is still in play). If the scales change, we know one of the two we swapped is the counterfeit. If the scales are the same, we know the two we swapped are legitimate (but we would have six suspects left, and only one weighing).

If we want to get maximum information out of the second weighing and solve this puzzle, we need to combine both tactics in our second weighing. Before our second weighing, we should swap one of the lighter side's coins with one of the heavier, then replace the three remaining heavy coins with legitimate ones. This can only produce three results:

• The lighter group is now heavier - this could only happen if the counterfeit was one of the coins we swapped. Weigh one of them against a known legitimate, if the scales balance then the other one is the counterfeit. If they don't, you've found the counterfeit.
  
• The lighter group is still lighter - this could only happen if the counterfeit is lighter than a legitimate coin and was amongst the three coins originally on the lighter side. You now have three suspect coins and know the counterfeit is lighter. Repeat the final step from the nine coins problem.
• The sides balance - this could only happen if the counterfeit is heavier than a legitimate coin and was amongst the three coins removed from the heavier side. You now have three suspect coins and know the counterfeit is heavier. Repeat the final step from the nine coins problem.