How many weighings of a balance are necessary to determine if a coin is counterfeit among eight coins. The counterfeit coin is either heavier or lighter than the other coins.
I understand the reasoning behind this problem when you know how the weight of the counterfeit coin compares to the rest of the pile, but I can not think of how to show that this problem takes 3 weighings.
What I have so far is that say you have the coins $A B C D E F G H$, weigh $ABC$ against $DEF$ if they are equal then weigh $A$ against $G$ if these are equal then the counterfeit coin is coin $H$ if these two are unbalanced then $G$ is the counterfeit coin.
Now this is where I hit the wall. When the piles of $ABC$ and $DEF$ are unbalanced you know that $G$ and $H$ are not counterfeit. I think you can now weigh $DA$ vs $BC$ and in the case where $DA$ and $BC$ equal you know that $E$ or $F$ is the counterfeit coin, which can be determined with one more weighing.
I am unsure of where to go from here, any ideas?