This completely ignores the human element, in both the act of creating and the act of solving a puzzle. There are two distinct methods of finding the correct piece: shape, and colour. A puzzle with regular pieces forces a potential solver to rely on colour and image characteristics such as lines, but is also utterly predictable. An irregularly carved puzzle need not, but it does have the potential for deliberate trickery.

Let me tell you about my dear grandmother. She's a puzzle-maker by trade, and she wields a mean jigsaw. She invariably sends us at least one of her puzzles for Christmas. Fiendishly difficult, they are quite idiosyncratic with distinctively loopy pieces and full of cunning traps. Because each piece is hand-cut on a jigsaw, no two are alike. Oftentimes, she'll carve a piece along a line of colour to make the neighboring one hard to find. I have also done puzzles of hers which seemed to have 5 corner pieces (one was a dummy), 3 corner pieces (one was disguised), and NO corner pieces (the puzzle, it transpired, was round). Algorithm THAT, Major!

Another thing to consider is that while the pieces themselves are not identical in an irregular puzzle, the elements that make up a piece (loops, curves, hooks, etc.) can still be quite quite common. Since the human eye is bad at gauging exact tolerances, the number of potential "fits" is astronomical. Furthermore, a non-regular piece gives no clue as to its correct orientation, so it may potentially fit in a much larger number of places. Putting constraints on the way a puzzle is made, as is is done with a regular puzzle, cannot possibly INCREASE its potential complexity, can it?

Really, having done both regular and irregular puzzles, I can personally testify that irregular puzzles are both considerably harder and much more engaging (though you are of course free to disagree). Irregular puzzles are a challenge, a gauntlet cast by the puzzle maker. If you've never done an irregular puzzle, I know a very good vendor in the New England area...

Also, do I see the word "pimp" in your writeup? Why yes, I think I do. Five times, no less.