I found two eight-move solutions to entropy.  Second is a slight variation of the first.

Number the board as

 1  2  3  4
 5  6  7  8
 9 10 11 12
13 14 15 16

The solutions are

6 2                and              6 2
10 6                                7 6
11 15                               11 15
7 3 4                               10 11 3 4
6 8 16                              6 8 16
15 3                                15 3
2 1                                 2 1
3 15 13                             3 15 13

where a line reads: move from first number - by way of any middle numbers - to the final number.

I suspect that you have seen these already - I did not find any more.  Found these by working backward - much easier than going forward.  Given time, might run a program to find all 8, 9, etc move solutions.

Very cute idea.