A bigger maze isn't a harder maze.

Get the red car out.

Cars slide along their own axis only — horizontal cars left and right, vertical cars up and down. Nothing jumps, nothing rotates. The red car escapes through the gap on the right edge.

Drag the cars. This one takes six moves.

Every puzzle contains a hidden map.

From the starting position you might have eight legal moves. Each leads to a position with its own eight. Follow that outward and you get every arrangement the puzzle can ever reach — its state space.

That number is the puzzle's size. It's the first thing anyone measures, because it feels like it should mean something: a puzzle with 11,000 possible positions has to be harder than one with 400.

Here's every puzzle in the set, sorted by size. The scale is logarithmic — each step right is ten times bigger.

Size is a liar.

First, what “difficulty” means here: the fewest moves a perfect solver needs. More moves, harder puzzle. It's a proxy for human struggle, not the same thing — but it's objective, and we can compute it exactly for every puzzle.

So compare two real puzzles. The left one can reach 11,487 positions; the right one only 422 — twenty-seven times smaller. If size drove difficulty, the left should be far harder.

That's not a cherry-picked pair. Across all 14,163 puzzles, size and difficulty are statistically unrelated. Flip the axis below and watch a shapeless cloud snap into a line.

What has to move before what.

When a car moves into space another car just left, the second one depended on the first. Trace those hand-offs and every puzzle reveals a structure.

Two real puzzles below. Both take exactly six moves — by solution length they're identical. Step them together and watch the hand-offs form.

Now the same measurement on two puzzles that are identical in length — both exactly 11 moves. One has a depth of 3. The other, 11: every single move is a prerequisite for the next.

Sometimes you have to drive away from the exit.

The instinct is to shrink the gap between the red car and the exit on every move. In hard puzzles that instinct is a trap: the winning move often makes the position look worse.

We count the fraction of moves in the optimal solution that fail to make greedy progress. In the puzzle below, 9 of 11 moves — 82% — go the "wrong" way. Watch the distance meter climb.

So we changed the shape of the world.

If dependency depth really governs difficulty, it should hold on a board that isn't a square grid at all. So we built one: a hexagonal board where pieces slide along three axes instead of two.

Nobody, as far as we can tell, has run Rush Hour on a hex grid before. It's not a reskin — the third axis genuinely changes the game.

Here's a real hex puzzle from the corpus — same rules, same red-piece-to-exit goal, but pieces now slide along whichever of the three axes they're laid on.

The third axis drains the tension.

More freedom means fewer traps. Hex puzzles solve in 4.3 to 7.5 moves on average, against square's 12.2 — and their dependency chains are shorter at every density we tried. Give pieces an escape route and gridlock stops being gridlock.

But pack the board tighter and structure reasserts itself. As hex density rises, dependency depth predicts difficulty better and better — 0.66, then 0.76, then 0.83. Crowding is what manufactures depth.

The scoreboard.

Five metrics, four board types, 14,163 puzzles. Each dot is how strongly that metric tracked difficulty on one board type — further right means a stronger relationship.