Here is the cabinet with new labels that accurately reflect the number of coins in each compartment:

Here again is the original arrangement, which serves as a basis for many of the clues given.

Clue #6 provides a good starting point for this puzzle.

6. Following one of the two diagonals: one compartment now has nine less coins, the next has ten less, the next has twice as many as it started with, the last has nine more.

It turns out that only one of the four possible diagonals mentioned in clue #6 produces an acceptable result.

Thus, going from the bottom right to top left, the compartments originally holding 16, 11, 6, and 1 coins now have 7 (nine less), 1 (ten less), 12 (twice as many), and 10 coins (nine more).

7. The compartment now containing six coins lies directly below the compartment which used to contain six coins.

Clue #7 puts 6 coins in the compartment which used to hold 10 coins.

4. One of the compartments now has half as many coins as it had before. It is directly below a compartment which now has twice as many coins as before, and above a compartment which has one less coin than before.

The next useful clue is clue #4, which addresses three compartments in a column. It cannot be applied to any of the compartments in the left-most column, or the column second from the right, since these contain odd numbers which cannot be evenly halved.

It also cannot be applied to any of the compartments in the column second from the left, since the middle two compartments have already been determined, and neither is half its original value (12 is not half of 6, 6 is not half of 10).

It cannot be applied to the bottom three compartments in the right-most column, since 7 is not one less than 16.

This leaves the top three compartments in the right-most column. Those will change to 8 (twice as many), 4 (half as many), and 11 (one less).

2. The new number of coins in one compartment is the square of the old number, and it is directly above a compartment with twice as many coins in it now than it used to have.

Next, clue #2 states that one compartment now has the square of the original number of coins it had. This can only apply to the compartments that originally held 1, 2, 3, or 4 coins. Original compartments 1 and 4 are filled, and the square of 2 (4) has already been used.

So the original compartment with 3 coins now has 9. Compartment 7 below it now has 14 coins (twice as many).

1. The compartment with three coins now lies directly above the compartment which now has sixteen coins.
5. The compartment which now holds thirteen coins lies directly to the right of the compartment which now holds fifteen coins.
3. The compartment which now has two coins is not in the left-most column.

Clue #1 arranges two compartments vertically that will have to be placed in the left-most column, while clue #5 arranges two compartments horizontally that will have to be placed in the bottom-most row.

There are three ways that these pairs can be arranged (either 16, or 15, or neither of those two is in the bottom left-hand corner). If the 15/13 set is on the right, then the 3/16 set can be put in one of two positions in the left-hand column. Unfortunately, in either position, the compartment with 2 coins has no place to go. Clue #3 says that the compartment with 2 coins is not in the left-hand column, and it cannot go in the same place as it was originally, since all the compartments must have a new number of coins.

Thus the 15/13 set must be placed all the way to the left in the bottom row, forcing the 3/16 set up into the middle of the left-most column.

Since the compartment with 2 coins cannot go back into the same place it was originally, it goes into the original 15 coin compartment. The 5 coin compartment then must go into the original 2 coin slot.