BT 5: Scales III

You have a tip-scale. You have N coins, one of which is counterfeit. You must identify the counterfeit coin using the scale no more than three times.

Your challenge is to determine the following: how large can N be? Note: there are four different scenarios to consider; you can get (at least potentially) four different answers.

Scenario A.1: You know one of the coins is counterfeit and that it weighs just a bit more than a true coin. You have a large supply of known-good coins at your disposal, in addition to the N possibly-bad coins.

Scenario A.2: You know one of the coins is counterfeit and that it weighs just a bit more than a true coin. You have no "extra" known-good coins to use in the process (i.e. all you have at your disposal are the N coins).

Scenario B.1: You know one of the coins is counterfeit, but you don't know whether it weighs more or less than a true coin. You have a large supply of known-good coins at your disposal, in addition to the N possibly-bad coins.

Scenario B.2: You know one of the coins is counterfeit, but you don't know whether it weighs more or less than a true coin. You have no "extra" known-good coins to use in the process (i.e. all you have at your disposal are the N coins).

Have fun,
Prof. Jacob