What makes a word puzzle hard, measured

A word puzzle is hard for reasons you can count, and for a long time we were guessing at which ones mattered.

The occasion was five levels in our middle hundred that were losing players at two to four times the rate of the levels either side of them. Not a smooth ramp with a few bumps — five distinct walls. Something about those boards was different, and "they feel harder" is not a thing you can act on.

Three ingredients, one equation

So we fitted median solve time against the shape of the board across about a hundred levels. Three things carried almost all of it.

Each extra answer on the board costs roughly 17% more time. Not 17% more per word in a linear sense — the median time for the whole level goes up by about that much for each additional word you have to find. Ten answers is a meaningfully different puzzle from eight.

Each three-letter answer gives about 5% back. Short words are nearly free. People find them almost immediately, and they also unlock crossings that make the longer words easier.

Each answer below a commonness threshold costs about 5%. We use Zipf frequency, where roughly 4.0 is the line between a word everyone produces on demand and a word people recognise but have to dig for. Below that line, every answer is a small tax.

Those three together explained the five walls without anything else being wrong. No bad clue, no layout bug, no broken wheel. The boards were simply carrying more words, longer words, and rarer words than their neighbours, and the effects stacked.

The obvious fix is the one you must not take

Look at that equation and the cheapest lever jumps out. If each answer costs 17%, take answers off the board. A ten-word level becomes an eight-word level and you have bought back a third of the solve time without touching anything else.

In our game, and in most of this genre, there are two tiers of word. Required answers fill the grid and you must find them all to finish. Bonus words are real words the letters can spell that are not on the board; find one and the game acknowledges it, but it is not part of the puzzle.

So "take a word off the board" really means "move it from required to bonus". The word still works. Nothing breaks. The level gets shorter.

We are not allowed to do it, and the rule is absolute: you may never move a common word to bonus.

Why that rule exists

Picture the player. The wheel spells BAT. They can see that it spells BAT. There is a three-letter row on the grid and they swipe BAT into it.

If BAT has been demoted to bonus, the grid does not take it. The row stays empty. From where they are sitting the game has just refused an obvious English word, and the fact that a scoring system somewhere acknowledged it does not help, because they were not trying to score — they were trying to fill that row.

You have not made the level easier. You have made it feel broken, and you have done it precisely at the moment the player was right.

This is the same failure we wrote about in our daily puzzle leaving out words, arriving by a different road. There it was a generator trimming a list. Here it would be us doing it deliberately, to hit a difficulty target. The player cannot tell the difference and should not have to.

So you change the wheel, not the word list

If a wheel spells more common words than the board should hold, the wheel is wrong for that slot. Pick a different one.

That sounds like a small distinction and it is actually the whole discipline. The word list is downstream of the wheel: once you have chosen the letters, the set of obvious words is fixed and you do not get a vote on it. All the design freedom lives one step earlier, in which letters you offer.

What you may do on a wheel you are keeping is move the genuinely rare answers down. A word below the commonness line is one nobody will swipe by reflex, so demoting it costs the player nothing and buys back about 5%. That is the legitimate version of the same lever, and it is much weaker, which is appropriate.

What the five fixes looked like

Four of the five walls were fixed by changing the wheel outright, to one whose obvious set was smaller. The fifth kept its wheel and lost two rare answers to bonus plus gained a three-letter one, which is the full toolkit applied to a single board.

One of them went from a nine-letter theme word to a six-letter one with seven answers. Another dropped from ten answers to eight. The equation says those should be substantially faster levels, and that is the prediction we shipped.

I want to be careful here, because this is where write-ups usually overclaim. We have not re-measured yet. The change is on a branch, it rides the next release, and the honest status is that we have a fitted model and a hypothesis, not a result. The model explained five walls we did not design; whether moving the inputs moves the outcome is a different question and a weaker one.

What the equation does not know

It was fitted on one stretch of one game's levels, so the coefficients are ours, not yours. The shape is probably transferable; the numbers are not.

It also has nothing to say about several things that obviously matter. Theme helps — a board where every answer belongs to one idea is easier than a random set, and we do not model that at all. Letter overlap matters: two answers sharing a rare letter are easier together than apart. And a word's grid position changes everything, because an answer with three crossings is nearly free by the time you reach it while an isolated one is not.

Most of all it cannot see the player. Someone on their fortieth level has a different sense of what is obvious from someone on their fourth, and a single median hides that completely.

If you are tuning your own

The useful part is not our coefficients. It is that difficulty in this genre decomposes into countable things, so you can fit it in an afternoon with data you already have, and then you can stop arguing about whether a level feels hard.

And when the fit hands you a lever, check what pulling it looks like from inside the puzzle. Ours said to take words off the board. From a spreadsheet that is a 17% saving. From the player's chair it is the game refusing a word they can see.