Can you solve it?

I had that toy when I was very young! Took ages to get it right!

This one is famously impossible.
The number of moves has to be even for the empty space to end up on the same place, but it has to be odd to only swap two blocks.

P12019
>This one is famously impossible.
how? I remembered when I first got that toy (the wooden ones and way smaller than the one in the picture. So small that it fits in your waistcoat pockets), I would spend the whole day getting the numbers in order 1,2,3,4,5,6,7,8,9,10,11,12,13,14

and I did it! when I got those number blocks sorted, I would mess it up and start fresh.

It was fun as you have to move those number blocks until you get them right, you can't remove those blocks. I love how smooth it was when flicking those number blocks with your thumbs

P12020
And if some cruel prankster were to take your toy, disassemble it, swap exactly two number blocks as shown in the OP picture, and put everything back together, then it would be rendered an impossible puzzle.

P12030 and if that happened, I would cry and suck my thumb as usual.

P12019
This one made me realize I knew about even and odd permutations but didn't know how to prove elegantly that even permutations could only be done in an even number of swaps and odd permutations only in an odd number of swaps. Best I thought of myself was decomposing into cycles and doing a case analysis, which works but is not elegant.

Apparently the easy way of seeing it is by looking at whether the inversion number is even or odd, where the inversion number means how many pairs of elements in the list are in reverse order.

In this puzzle, if we consider the empty square to be a 16, we can count the number of pairs of squares which are in reverse order. In the OP picture there is exactly one such pair, the 15 and 14, so the inversion number is 1, an odd number. In the desired state, the inversion number is 0, an even number. Each move changes the inversion number from even to odd or vice versa, which is why the number of moves to swap only two blocks must be odd.

What's a fast way to look at a scrambled 15-puzzle and determine whether it is solvable or not?