Modeling what makes paper-folding puzzles hard

You've probably done this as a kid. Take a piece of paper, fold it a couple of times, punch a hole through the layers, then unfold it and see where all the holes ended up. It's fun. It's also, as it turns out, a standardized cognitive test.

The paper folding test shows up in real psychometric testing. The most direct version is Ekstrom et al.'s VZ-2 “Paper Folding Test” from their 1976 Kit of Factor-Referenced Cognitive Tests. It measures spatial visualization, your ability to mentally manipulate 2D objects through sequences of transformations. When I started building, I assumed difficulty would scale with grid size and fold count. After a lot of playtesting, that turned out to be only part of the story.

I turned this into Daily Unfold, a daily puzzle game. Three difficulties each day: easy is a 4x4 grid with one fold, medium is 6x6 with two folds, hard is 6x6 with three folds and two punch holes. The game generates puzzles deterministically from the date, so everyone in the world gets the same three puzzles. But making that difficulty curve feel right? That was the real problem.

Simulating the folds

The core engine runs a forward simulation. For every cell in the original unfolded grid, it traces that cell through each fold one by one. If a cell is on the folding side, its coordinate gets mirrored across the fold line:

// For a horizontal fold at position p:
if (row <= p) {
  row = 2 * p + 1 - row;  // mirror
}
row = row - (p + 1);        // shift to new origin

The smaller side always folds onto the larger side. After mirroring, coordinates get shifted so the remaining visible portion starts at zero. Each fold shrinks the effective grid, and the next fold operates on whatever is left.

Once all folds are applied, the engine checks if the cell's final position matches any punch location. If so, that cell has a hole in the unfolded paper. With three folds, a single punch can go through up to 2³ = 8 layers, producing up to 8 holes.

A center fold creates a clean mirror. The punch at (2,1) maps to (1,1) on the other side.

How I model difficulty

My first approach was obvious: bigger grid, more folds, harder puzzle. That produced a lot of puzzles that felt wrong. A 6x6 with two center folds could feel trivial, while a 4x4 with one weird fold stumped people. I needed something better.

After reading some spatial reasoning research and doing a lot of playtesting, I landed on a scoring function that weights six factors. The weights are hand-tuned, not learned from data. Here they are, ordered by weight:

The general pattern: factors that disrupt mental shortcuts (off-center folds, scattered holes) got higher weights than factors that just add more stuff to process (hole count, grid size). The scoring function cares more about how a puzzle is structured than how much stuff is in it.

score = holes      x 1.0
      + folds      x 2.0
      + offCenter  x 4.0
      + mixedAxes  x 3.0
      + punches    x 2.5
      + spread     x 4.0
      + gridBonus  x 2.0

Generating puzzles that feel right

Every puzzle is generated deterministically from the date. Today's hard puzzle comes from hashing the string UNFOLD_HARD_2026-03-25. Same date, same seed, same puzzle, no matter what device you're on.

But random generation doesn't always produce good puzzles. Sometimes a “hard” puzzle rolls lucky folds and ends up feeling easier than the medium. Sometimes the hole count is boring. So there's a validation step after generation.

After generating all three puzzles for the day, the engine checks that the cognitive scores are properly ordered: hard > medium > easy. If something is out of order, it regenerates the offending puzzle with an offset seed: UNFOLD_HARD_2026-03-25_v1, then _v2, up to 10 retries per difficulty.

The rerolling itself is deterministic. The version suffix is part of the seed, so the search for a good puzzle follows the exact same path every time. No server, no database. Just a hash function and a PRNG.

// The full generation pipeline
const seed = hash("UNFOLD_HARD_2026-03-25");
const rng  = new SeededRandom(seed);

let puzzle = generate(rng, "hard");
let score  = cognitiveScore(puzzle);

// Reroll if the difficulty curve is wrong
for (let v = 1; score < mediumScore && v <= 10; v++) {
  const offsetSeed = hash("UNFOLD_HARD_2026-03-25_v" + v);
  puzzle = generate(new SeededRandom(offsetSeed), "hard");
  score  = cognitiveScore(puzzle);
}

Hard puzzles also have structural constraints baked in. The engine guarantees at least one off-center fold (to break symmetry) and folds on both axes (to require 2D reasoning). If the random generation doesn't satisfy these, it mutates the folds, flipping an axis or shifting a position, then revalidates.

Where this falls apart

The weights are hand-tuned. There's no regression, no training data, no formal validation. The spatial cognition literature gave me a starting direction (asymmetry and mixed-axis transformations seem disproportionately hard), and playtesting gave me the specific numbers.

There are definitely puzzles the model scores as “medium” that players find brutal. This usually happens when the fold sequence produces a hole pattern that doesn't match any simple mental template. Template-matching (when you recognize a pattern and skip the reasoning entirely) is a cognitive strategy the model doesn't capture at all.

It also ignores practice effects. Someone who has played 50 days builds fold-specific intuitions that a new player doesn't have. What feels hard on day 1 might be routine by day 30. You could build an adaptive system that adjusts to individual skill, but that breaks the “everyone gets the same puzzle” property that makes daily games fun to share.

With enough data (thousands of solves with timing), you could train a real model. Maybe a regression on solve time, or something that predicts error patterns per fold configuration. For now the hand-tuned heuristic gets the job done. The difficulty curve feels smooth for most players, and that's what matters.

One PRNG, no server

The whole thing runs client-side. No server generates puzzles, no database stores them. A date string gets hashed into a 32-bit integer, which seeds a mulberry32 PRNG. It's tiny (8 lines of JavaScript), fast, and has good distribution. Every random decision during generation (axis choice, fold position, punch location) comes from this one sequence.

The result is a fully deterministic system. Same date, same puzzles, every device, every browser, no network request. The rerolling mechanism works because appending _v1 to the seed string doesn't retry randomly. It follows a specific, reproducible alternative path through the generation space.

Further reading

If spatial cognition interests you:

• Ekstrom et al. (1976) – Kit of Factor-Referenced Cognitive Tests. The VZ-2 “Paper Folding Test” is essentially this game. It's the direct ancestor and the reason I knew the task was a real measure of spatial ability.
• Shepard & Metzler (1971) – Mental Rotation of Three-Dimensional Objects. Not about paper folding specifically, but the foundational work on mental spatial transformations. Their finding that rotation time scales linearly with angle is what got me thinking about fold count as a linear cost factor.