It is not the number of pieces, and it is not the subject. It is how much the picture repeats itself.
Ask most people what makes a jigsaw difficult and they will say the number of pieces. Ask a puzzler and they will say the sky.
Both are right, but only one of them is interesting. Piece count sets how long a puzzle takes; the picture sets whether it is enjoyable. And the property that governs it is not the subject at all — it is how much neighbouring parts of the image differ from each other.
A jigsaw is solvable because pieces carry information. A piece with a bright edge, a change of colour, a line running through it, tells you roughly where it belongs before you have tried it anywhere. A piece cut from an unbroken field of one colour tells you nothing, and the only way to place it is to try it against every gap until one fits.
So the hard parts of a picture are the uniform parts. A tiger is easy despite being one animal, because stripes are effectively a map. A forest floor is easy despite looking chaotic, because chaos is information. A clear blue sky across the top of a landscape is where the difficulty of that puzzle actually lives, and it is why experienced puzzlers leave it for last.
Every picture in the library is graded automatically rather than by hand, by measuring how much each part of the image differs from the parts next to it. Pictures with a lot of local variation come out easy; pictures with large flat regions come out hard. Nobody sits and judges whether cathedrals are harder than cats.
The result is sometimes counterintuitive, and that is the point. A busy city scene often grades easier than a minimal landscape, even though the city looks more complicated. Complicated is not the same as hard. What you are looking for is whether two pieces from different places could plausibly be swapped without you noticing — and in a photograph of a city, they could not.
Turning on random rotation means every piece arrives at one of four angles, and you can no longer use the shape of a piece to rule out a gap. It is roughly a doubling of difficulty on any picture, and it works on easy pictures too — which makes it the better way to make a simple image harder than piling on pieces.
If you want to test any of this, the puzzle of the day is the same picture and the same cut for everyone, so times are directly comparable — which is the closest thing to a controlled experiment a jigsaw offers.