Start with a solution, then make the challenge
The generator begins with a square and a candidate sequence of straight folds. It applies those moves in order and uses the resulting paper footprint as a possible target. A retained sequence can be replayed from the original square. The puzzle therefore comes with evidence of reachability rather than an assumption based on how its outline looks.
That distinction rules out a misleading shortcut: choosing an arbitrary polygon and hoping players find a way to create it. A plausible drawing is not a proof. Even a simple outline can be unreachable under a particular set of directions or board guides. The sequence must work under the same rules that the player uses.
Represent paper as layers of polygons
At the start, the sheet is one square polygon. A crease divides the plane into two half-planes. The engine clips each current layer against that line, separating the piece that stays from the piece that moves. Clipping inserts an intersection point wherever a polygon edge crosses the crease, preserving the boundary of each piece.
The moving piece is reflected across the crease and placed over the stationary piece. Repeating this operation records overlapping layers instead of deleting excess material. The visible footprint is the area covered by at least one layer. Keeping those concepts separate matters: a smaller footprint does not mean the paper has lost material.
Use the same reflection for every point
The engine describes a crease as n · p = c, where n is a nonzero normal vector, p is a point, and c fixes the line’s position. It reflects a point using p′ = p − 2(n · p − c)n / (n · n). Points on the crease stay fixed; other points land the same perpendicular distance on its opposite side.
Dividing by n · n matters because the diagonal normal (1, 1) is longer than the vertical normal (1, 0). The formula handles both consistently. It also explains the midpoint rule used while playing: a crease sits halfway between a point and its reflected destination. Dragging and the precise selectors share this transformation.
Reject folds that do not make a useful puzzle
A mathematically defined reflection is not automatically a useful move. A crease outside the current paper changes nothing. A line that only touches one corner can create numerical fragments without producing a meaningful fold. A candidate sequence also needs enough visible paper for players to see and control what remains.
Useful validation checks therefore include actual paper on both sides of the crease, nonzero polygon area, finite coordinates, and a target that differs from the starting square. Replaying a retained sequence checks that it still reaches the saved target. These checks address the software model; they should not be described as a full laboratory test of physical paper.
Understand what the model promises
Crease Game models flat, straight folds that move all affected layers together. It does not simulate paper thickness, friction, elastic bending, or every intermediate three-dimensional movement of a real sheet. It also does not require specialist origami maneuvers such as curved creases or selectively pulling an interior layer through the stack.
Reachability here means that the permitted sequence produces the target when executed by the game’s folding engine. That is a narrower, inspectable claim than saying every possible real material will behave identically. If you try a puzzle with paper at home, use a square and follow the same crease positions while allowing for ordinary material thickness.
A solution route is different from a minimum proof
Saving a working sequence proves that a target can be reached in that many moves. It does not, by itself, prove that no shorter route exists. A player may discover a clever overlap that improves on the generator’s route. The displayed Fold goal uses the recorded solution length; the game does not claim that it is a proven minimum.
For players, the practical benefit is straightforward: every published target should have a reproducible way forward. You can experiment, undo, and ask for a hint with a concrete destination to pursue. The challenge comes from recognizing the folding relationships, while the generator’s job is to make sure those relationships lead somewhere.