Difficulty depends on
who is playing

CLEAR RATE BY LEVEL

Ten times the training moved the agent from 56% to 59%. A twenty-line greedy rule, which simply takes whichever swap clears the most target tiles right now, still reaches 83%. That gap is the honest result: PPO learns, more training helps, and in this environment a one-move heuristic is still ahead of it.

Watch the gap

Every player below is dealt the identical starting board, so what differs is the choices, not the luck. Averages hide what that looks like; the boards do not. These three deals are picked rather than drawn — an arbitrary seed gave three levels where everyone won or everyone lost, which illustrates nothing. The curves above are the measurement; these are one episode inside them.

Which levels reward playing well

Subtracting the random player’s clear rate from the greedy player’s gives a number per level: how much skill is worth there. It peaks at 63% on level 27. On 8 levels it is under five points — those levels cannot tell a careful player from a careless one, which is worth knowing before shipping them as a tutorial.

The walls that were not there

A spike detector looks for a level-to-level drop too large to be chance. My first version used one fixed threshold for every curve, and flagged three walls on the greedy curve. They were measured with sixty episodes against the others’ two hundred; re-measured with five times as many, those drops fell from twenty-odd points to single digits. They were never levels. They were the sample size.

The threshold now scales with the measurement behind it — 3 standard errors of a difference in proportions, so a curve built from fewer episodes has to clear a higher bar.

2 drops clear the threshold. The level table was built as a straight line, so anything flagged is worth opening before it is believed.

The PPO is mine, and checked

Every number above rests on an agent, so the algorithm behind it had to be verified rather than trusted. I wrote PPO out — actor and critic, advantage estimation, the clipped objective, action masking — and trained it on the same environment as stable-baselines3, for the same number of steps, from the same seed.

The verdict was decided before the run: the two agree if the mean gap between the curves is smaller than the spread of the reference curve itself. It was.