How to programmatically solve the 15 (moving numbers) puzzle?

Question

all of you have probably seen the moving number/picture puzzle. The one where you have numbers from 1 to 15 in a 4x4 grid, and are trying to get them from random starting positi

Answer 1

By reduction the only possible case you can't solve must be of the form
1 3
2 X
and you want to get it to
1 2
3 X

by using an additional row and column you can move those to the proper positions with a simple precomputed sequence