What easy, medium and hard mean here

The word under the board on any given day — easy, medium, hard — is not an opinion about how the puzzle feels. It is decided before the puzzle exists, it fixes two numbers, and then a separate measurement decides whether the puzzle that came out is allowed to be that day's puzzle at all. Those are three different things and it is worth keeping them apart.

The label is the weekday

Difficulty is a lookup on the day of the week, the same for everyone:

• Monday and Tuesday: easy.
• Wednesday and Thursday: medium.
• Friday and Saturday: hard.
• Sunday: medium.

The weekday is computed in Jerusalem, which is where the studio is. The Friday-and-Saturday choice is deliberate: the site is read in both kinds of week, and in Israel Sunday is a working day while Friday is not. Hard at the end of the Gregorian week is also the weekend for the Israeli one, so the same curve serves both.

Nothing about you enters this. The puzzle does not get easier because you lost yesterday, and it does not get harder because you are quick. It is generated up to a month before you see it.

What the label changes

The label sets exactly two things: how many moves you must play, and how many tiles you get.

Read that table honestly and something looks wrong. An easy board offers ten possible three-tile sets out of five tiles; a medium board offers only five four-tile sets out of five; a hard board offers six. By that count medium is the narrowest of the three. What actually grows is everything else: the chain is longer, so the running value travels further and the arithmetic gets bigger; more of your hand is in play at once; and the number of distinct legal sequences you could stumble through goes from 85 on an easy board to 205 on a medium one to 1,236 on a hard one. Fewer sets, far more ways to be inside one and still lost.

The score is measured, not guessed

Once a candidate puzzle has been generated and completely solved, it is scored. The score never looks at the move count — it looks only at features of the solved board, and it exists to answer one question: is this a typical puzzle for its length, or a freak?

The formula, in full:

• Branching. At every step along the solution, count the unused tiles that are legal from that position, add them up, and subtract the number of steps. What is left is the number of wrong turns that were available to you.
• Near misses. Full-length chains that end within ten of the target without hitting it. Counted, then capped at six.
• Decoy strength. How many spare tiles are still playable at the second-to-last move — spares that never rule themselves out for you.
• Plus three for a divide in the winning set, and plus three for swap digits.
• Minus two for every extra ordering of the winning set that also reaches the target, capped at three extras. If six orders work, you only had to find the set.

Multiplication earns no bonus, and that is not an oversight. Divide and swap are the two tiles that can be refused: a divide only plays when the value is a multiple, and a swap only plays when it would change the value. A tile that is sometimes illegal is a tile whose position in the chain you have to think about. A multiplier will always go.

Where the bands came from

The bands were not chosen. They were sampled. A calibration run generates a thousand candidate puzzles for each solution length with the band check switched off, scores all of them, and takes the 30th and 70th percentile of the resulting distribution — the middle 40%, with both tails discarded.

A candidate that scores outside its band is discarded and the generator tries again with the next attempt in its stream. About a third of the candidates that get as far as this check die here — 31 of the 91 that reached it over a recent sixty-day run. That is the point: it throws away the freakishly gentle three-move puzzle and the five-move monster alike, so that two hard Fridays feel like each other.

Notice that easy and medium share a band. Their difference really is the length. Hard sits above both, so a hard day is not merely longer — it is denser in the features above as well.

Those numbers are regenerated by a script, written into the engine as a generated file, and pinned to an engine version, so re-tuning them can never quietly change the difficulty of days that already exist on the archive.

Three real days, scored

These are the puzzles already worked through in how to play, with their actual numbers.

Monday 7 September, easy — 79 to 160 in three, hand +65 −28 +55 +54 −36. Legal tiles at each step: 5, then 4, then 3, which is 12; minus three steps gives 9 of branching. There are 24 near misses, capped to 6. Both spares are still playable at the second-to-last move, so decoy strength is 2. No divide, no swap. All six orderings of +54 +55 −28 reach 160, which is the maximum discount, minus 6. Total: 9 + 6 + 2 − 6 = 11 — right on the floor of its band.

Sunday 13 September, medium — 41 to 386 in four, hand +15 swap digits +92 +40 +14. Branching 10, four near misses (all of them landing on 385), one live spare, plus 3 because the winning set uses swap digits, and four valid orderings costing the full 6. Total: 10 + 4 + 1 + 3 − 6 = 12.

Friday 11 September, hard — 21 to 200 in five, hand −37 +70 ÷3 +82 +84 +76. Legal tiles per step 5, 4, 4, 3, 2 — eighteen in total, minus five steps, so 13 of branching, the highest of the three. Thirty-six full-length chains end on 198 or 206, capped to 6. One live spare. Plus 3 for the ÷3 in the winning set. Eighteen orderings of the right five tiles reach 200, so minus 6 again. Total: 13 + 6 + 1 + 3 − 6 = 17 — which is exactly the median of the five-move distribution.

So what actually makes a day hard

Reading those three across, the concrete answers are these. Every extra move multiplies the sequences you can wander into. A divide or a swap in the winning set is worth three points on its own — more than half the width of the entire easy band — because it constrains when the tile can be played, not just whether it belongs. Near misses are the tax on checking: a hand where thirty-six different chains stop within ten of the target is a hand that will keep telling you that you were nearly right — which is why the strategy guide treats a near miss as evidence about one tile rather than about the order. And a spare that is still legal at your last decision has cost you nothing to carry and given you nothing to eliminate.

Pulling the other way is the ordering discount. A set that can be played eighteen different ways is a set you only have to identify, not sequence, which is why the hard puzzle above scores 17 rather than 23. When a puzzle has exactly one legal order, nothing is subtracted, and it will feel it.

What the difficulty label is not

It is not derived from how anyone did. No solve times, no completion rates and no live statistics feed back into it; the label and the score are both fixed before the first person opens the page.

It is also not the score. The number never leaves our server — your browser receives the start, the target, the hand and the move count, and that is all. What you see is the word and the row of pips telling you the budget.

And it does not vary by person. The Friday puzzle you are stuck on is the same Friday puzzle everyone else is stuck on. If you want a specific difficulty rather than today's, the archive lists every past day with its label, so a run of Mondays is three moves each, all the way back.