Rough idea: Both pick an opposite corner piece. That piece and any connected to it are yours, and locked to you. All other pieces are fair game, and can be rearranged by anyone - any player may claim and lock a piece (or clump of pieces) if it would connect to the claiming player’s locked pieces. If a piece would connect to either player’s section, whoever grabs it first gets it.

(So you can start building an easy middle-of-the-picture section even if it doesn’t connect to your home corner yet, but if your opponent reaches across far enough to touch that section before you do, they can claim the whole thing.) (Doesn’t seem likely that anyone would actually want to build an unconnected clump, but I don’t know. Perhaps more amusing as an agreed pre-built “power-up” to race to; one player chooses and builds a clump, the other player picks the first corner.)

The game is over when no more pieces can be locked - you should end up with a jigsaw that looks as if it’s been torn jaggedly in half. Whoever owns the majority of it is the winner.

Optional balancing rule: players have to take turns joining one piece (or clump) to another piece, making the game a bit more conversational-background and removing (some of) the advantage that a jigsaw-veteran would have.

Oh, wait, turn-taking makes it completely pointless, doesn’t it? Tsch.

Combine the jigsaw with a capture game or a wargame. On a given turn you can move units around, or build an additional piece of terrain. Units inhabit one of the jigsaw pieces and can move to adjacent pieces.

Excellent. With the picture fragments on the pieces giving your pawns different abilities while they’re there, perhaps, although you might have to tailor this to individual jigsaws.

“If there’s a bit of sky, you can move that pawn to any other bit of sky.” “If there’s a black bit larger than 2mm, you may remove any piece from the jigsaw instead of moving that pawn.” “If there’s an eye, you may hide a piece in your pocket.”

Which I suppose is leaning towards Zarcana a bit, if we want a specimen to analyse.

If the jigsaw depicts a landscape then it could work just like a regular wargame board with different types of terrain. You would have the very odd experience of having the sky being a specific place where only sky units could go, rather than a third dimension perpendicular to the board as in every other war I’ve seen. It would be like a cross between a wargame and a 1980s video game.

Possible system for instant tailoring to unique puzzles: Have a set of thirteen or so special properties, like Kevan lists above (”move pawn to like piece,” “remove a piece,” “claim a corner,” etc). When you first get out the puzzle, have everybody look at the box and list a few noticeable entities each (”the sky,” “sheep,” “blood spatter”). List these on a piece of paper. Assign the properties to one suit from a deck of cards, shuffle those cards and deal them out to match properties with features.

Obvious extension: might work better to have twenty properties and ten features, so there’s no guarantee any property will always be part of the game.

Each player takes a piece, one at a time. The player that ends with the largest contiguous mass of pieces wins.