Years ago, mathematicians fooled around with various coin weighing problems. Perhaps you folks might like to have a try at solving some of them.

The basic idea is that true and counterfeit coins can only be distinguished due to a difference in weight. All true coins have the same weight. All counterfeit coins have the same weight. In none of the problems do you have a true coin available for comparison purposes.

Here are three problems.

First problem.

You are given 12 coins, one of which is counterfeit, and a balance scale. In three weighings, determine which coin is counterfeit and determine whether it is lighter or heavier than the true coins.

Second problem.

An emperor has 15 kings who pay tribute in coins of the realm. There is a ceremony in which each king places a pile of coins in front of the emperor. One year, a spy tells the emperor that one of the kings is going to pay him in counterfeit coins. Before he can identify the culprit king, he is killed by an arrow from an unknown source.

Using a scale which will give the exact weight of any number of coins put on it, the emperor intends to start weighing one coin from each king's tribute. To identify the counterfeit pile of tribute, the weight of a true coin, and the weight of a counterfeit, he figures to use the scale at most 15 times. If lucky, he will use it only 3 times.

A court mathematician tells him the following.
Your plan might be the fastest method, but in theory you can always solve the problem in three weighings, assuming there are enough coins in each pile of tribute, and it looks to me as though there are enough coins for the solution I have in mind.
Can you think of the mathematician's solution to this problem?

Third problem.

Several years later: Same emperor with 15 kings paying tribute. This time the emperor is told that 5 of the fifteen kings are crooks paying in counterfeit coins. This time the mathematician says.
You should start weighing one coin from each pile. I do not think there are enough coins in each pile of tribute to solve the problem in 3 weighings, but in theory it could be done with really big piles of coins. In practice, no non-mathematician would consider the solution I have in mind., but we mathematicians enjoy thinking about cute theoretical solutions to such problems.
What did the mathematician have in mind?