What is PLICA — Fold the page until every seal meets its reflection
PLICA is a folding puzzle. The board is a sheet of paper on a dark desk, and the only move in the game is to drag one half of it over a crease. The half you drag travels, arrives mirrored, and — this is the whole game — everything printed on it comes back showing its other face: solid ink seen through the back of the page becomes hollow, and hollow becomes solid. Two seals of the same symbol that end up in the same square close for good when their faces disagree. Two faces that agree would smudge, so the page refuses the fold and springs back.
A run is twelve sheets, from a four-by-three page with two pairs to a six-by-six page with five pairs and two brass nails holding it to the desk. Every sheet is built by choosing the folds first and only then printing the seals on the squares those folds bring together, so a solution always exists; a breadth-first search then plays the sheet out to the end and returns the smallest number of folds there is. That number is the par printed beside the fold counter. The clock starts at six minutes and every sheet you close adds one more, so a run lasts exactly as long as your folding deserves.
How to play
- Drag across the page with a finger or the mouse. The half you start from folds over the crease nearest to where you touched, and lands mirrored on the other side. On a keyboard the arrow keys pick the direction the paper travels — press the same arrow again to shift the crease — and Enter makes the fold.
- Every fold turns the ink of the half that travels. A solid seal comes back hollow and a hollow one comes back solid, so the face of a seal depends on how many times it has been carried across a crease.
- Two seals of the same symbol that meet with opposite faces close and leave the page. Two with the same face are a blot: the fold is refused, nothing is lost, and you can try elsewhere. Close every seal on the sheet to move on.
- A brass nail pins its half of the page to the desk, so that half never travels — choose the other one. Undo costs fifteen points, a hint seventy, and matching the par is worth two hundred and fifty.
Strategies that work
- Count the flips, not the distance. Two seals printed with the same face need an odd number of folds that move exactly one of them; two with opposite faces need an even number. A fold that carries both of them across at once changes nothing between them, which is why an early fold of the whole page so often leaves the sheet exactly as hard as it was.
- The cheapest flip is the crease right next to the edge. Folding the first column onto the second moves a single column and leaves the rest of the page untouched: one seal changes face, everything else keeps its own. When a pair is one square apart with the same face, that little fold is usually the whole answer.
- Creases away from the middle make the page hang over the far edge, and that is legal. Folding a six-wide page at the fifth crease leaves it five wide, sticking out four columns past where it started. Two seals an odd number of squares apart can only be brought together by an off-centre fold like that one.
- Read the nails before anything else. A nail kills every fold that would carry its half, so a nail in the first column forbids all folds that send the left side rightwards, and a second nail in the last column would forbid the whole horizontal axis. When both ends of one axis are nailed, the answer lies entirely in the other.
- When the fold counter turns red you are already over par, and the sheet is still worth two hundred points plus whatever the clock gives you. If the page tells you it can no longer close, undo instead of restarting: fifteen points against a fresh sheet and the time it costs to think it through again.
A little history
Plica is the Latin word for a fold, and the puzzle is built on something anyone who has held a sheet of paper up to a window already knows: what is written on one side reads backwards from the other. Mathematicians call a shape that cannot be laid over its own mirror image chiral — a left hand and a right hand are the standard example — and every seal in this game is drawn to be chiral on purpose, so that a fold is impossible to miss. The rest follows from one honest constraint of paper: a fold moves half the sheet and leaves the other half exactly where it was.
Frequently asked questions
Can a sheet be impossible?
No sheet arrives impossible. Each one is generated by choosing a sequence of folds first and printing the pairs on the squares that sequence brings together, and a breadth-first search then proves the solution and measures the par. You can, however, fold yourself into a page that no longer closes — when that happens the game says so under the board, and undo takes you back.
Is the daily challenge the same for everyone?
Yes. In daily mode the twelve sheets come from the calendar date in UTC, never from the clock on your own computer and never from chance, so the page you are folding in Lisbon is the same page somebody is folding in Montevideo. The daily run counts once; after that you can keep playing freely.
Does it work on a phone?
Yes. The whole game is one drag, which is as comfortable with a thumb as with a mouse, and the page has been measured to fit a small screen without scrolling. There is nothing to install and nothing is loaded from another server.
