I made 2 videos about this: and

but here's the upshot: You have a 50/50 chance of the puzzle being solvable. Check the comments in the 1st video for the maths. But the fastest way to see if it's solvable or not, is trying to solve it, and it either breaks down in the three last steps, or it doesnt.

This Image has the outlines and the numbering: https://imgur.com/a/gJeypNg 1 2 3 4 5 6 7 8 9 10

Step 1: Find out which piece is missing. If it's 4 5 9 or 10, start by solving 1 6, then 2 7, the 3 8. If it's 1 2 6 or 7 that's missing, start with 5 10, then 4 9, then 3 8. If 3 or 8 are missing, solve the outsides first and end on the inside. If the remaining 3 pieces cannot be put in the correct place at the same time, it's impossible.

How to solve 1 6: Either: put 6 in top left corner, then 1 to the right of it. Then slide both in from below. Or: put 1 in the bottom left corner, then 6 to the right of it, and slide in from above.

Same goes for 2 7 as soon as you've fixed 1 6, and the same goes mirrored for 5 10 and 4 9.

Slightly faster strat that does not require finding out which piece is missing:

at least one of the pairs 1,6 or 5,10 must be on the screen, as only one tile can be missing. Therefore, find the pair that exists, and solve it. Repeat for the remaining 4x2 puzzle, and the final 3x2.