The king of Ebonchester has just returned from his recent conquests of Baronshire and with him he brought four cart loads of plunder. However Baronshire is known for its lacklustre fiscal regulation, so the king would like to test the authenticity of every coin. The problem is his old accountant died at his side in battle (a good king never goes anywhere without his accountant). Thus he wants to recruit the most meticulous counters in the land.
Having placed posters on every street corner offering the handsomely paid job he was soon inundated with applications. Each applicant had to prove their worth by determining the forged coin from a stack of 12 (which could either by lighter or heavier) in just 3 weighings.
Since the great infrastructure drive of a few years prior everyon in the kingdom has internet access, and it turns out that everyone and their mother had rushed to Stack Exchange in preparation for their interview and every single one of them knew the answer.
So the king went back to the drawing board and devised three new and more fiendish tests:
- There are 12 coins, of equal size and shape, and exactly 1 is slightly lighter or heavier
- There is a scale available that can tell you whether one side is heavier than another when coins are weighed against each other
- Each weighing may consist as many or as few coins on either side as possible
- The coins can not otherwise be determined apart
- The coins are labelled 1-12
- Any relabelling is done without you knowing which coins have had the label change
In the first test once the scale has fallen to the left or right during a weighing, it must fall the same way, or balance equally, in all future weighings.
After each weighing the number of the counterfeit coin is swapped with:
- The coin labelled one higher than its current number if it is lighter than the other coins
- The coin labelled one lower than its current number if it is heavier than the other coins
If the coin is light and numbered 12 it would swap with 1 and if it is 1 and heavy it would swap with 12. That is the number on a light coin increases by 1 each weighing (mod 12) and decreases on a heavy coin by 1 each weighing (mod 12)
As per test 2 but now a coin increases or decreases (mod 12), when light or heavy respectively, by the current number on the coin itself. For example if coin 7 was light, after the next weighing it would have its label switched with coin 2, and if it were heavy it would switch with 12.
Question: What is the minimum number of weighings required (if possible) to be certain of which coin is the counterfeit and whether it is light or heavy in each case?
Disclaimer: These are original variations, and while I have solutions of my own, I am not certain they are optimal for tests 2 or 3