What actually makes a puzzle hard

Size is barely relevant

A thirty-card Memory board is not much harder than a twenty-card one; it just takes longer. A fifteen-wide nonogram is not much harder than a ten-wide one, for the same reason. Meanwhile a six-crate Sokoban level in a tight room can defeat someone who finished the previous eleven levels comfortably.

Size affects duration. Difficulty is something else, and it comes from four properties that vary independently.

One: how many options each move offers

At any point, how many things could you reasonably do? In Hanoi, almost always one sensible move. In Connect Four, seven. In Sokoban, four directions times however many crates are reachable.

High branching makes a puzzle hard to think ahead in, because looking three moves forward means considering many more positions. It is also what makes a puzzle resist brute force: you cannot try everything if trying everything means thousands of sequences.

Notably, high branching does not make a puzzle hard to solve if there is a method that ignores most of the branches. The sliding puzzle has plenty of legal moves at any moment and is not difficult, because the layer method tells you which two or three matter.

Two: how far ahead you have to see

Depth is how many steps separate a decision from its consequence. This is the property most closely tied to the feeling of difficulty, because it decides how much you must hold in your head.

Sudoku is shallow at easy settings — each cell is decided by what is around it right now — and gets deeper as the techniques get longer. An X-wing is a chain of reasoning about four cells across two rows and two columns, and holding it in mind is most of the work.

Sokoban is the deepest puzzle here. A push on move four can make the level unsolvable in a way that only becomes visible on move thirty. That gap between cause and symptom is precisely why it is hard, and why planning backwards from the marks beats pushing forwards.

Depth is also what makes a puzzle rewarding rather than merely long. A deduction that spans six steps feels like an achievement; one that spans one feels like arithmetic.

Three: whether mistakes are recoverable

This is the property people underrate most, and it may be the single biggest factor.

In Sudoku, a wrong digit is a nuisance: you notice a contradiction, back it out, continue. In the sliding puzzle, any position can be reached from any other reachable position, so no move is ever fatal. Both puzzles let you explore, and exploring is how people learn.

In Sokoban, a crate pushed into a corner is finished. The level is over, and you may not find out for twenty moves. Irreversibility means you cannot learn by trying — you have to reason before acting, which is a completely different and much more demanding activity.

Four: whether certainty is available at all

Nine of the ten puzzles here can always be advanced by proof. If you cannot find the next step, it exists and you have missed it, because the generators guarantee it.

Minesweeper cannot make that promise. Some positions genuinely have no determined cell, and the correct play is to work out which cell is least likely to kill you. That is a different skill from deduction — it is estimation — and it is why the probability guide exists as a separate piece.

Connect Four is a third case: certainty exists but belongs to your opponent as much as you. The game is solved, so perfect play has a known outcome, but you are not playing against perfection and neither is the computer at the settings offered.

This property changes how a loss should be read. Losing a Minesweeper board may mean nothing at all about how well you played. Failing a Sokoban level always means something.

Why our difficulty labels are estimates

Some of this is measurable. We can count a Sudoku grid's givens and which techniques it requires. We can compute the minimum pushes for a crate level and the minimum moves for a Hanoi stack. Those are facts.

What we cannot measure is how hard something feels, and that is what a difficulty label claims to describe. It depends on which patterns you already recognise, and a grid needing a technique you know is trivial while a grid needing one you do not is impossible. The same board is both.

So every label on this site is our own judgement from our own play, marked as an estimate wherever it appears — as are the completion times. They are there for rough comparison between our boards, not as a claim about you.

The one number worth trusting is your own best time, kept in your browser and compared with nobody. It is the only measurement on the site with a properly controlled variable.