How puzzle difficulty is actually measured

Why the number of clues does not grade a sudoku, what does, and how a maze and a word search are measured instead.

The difficulty printed on a sheet is a promise, and it is the one thing a teacher cannot check before handing it out. A puzzle labelled hard that collapses in two minutes is worse than an unlabelled one, because the label is what made you choose it. Nearly every generator declares a level; the question worth asking of any of them is what was measured to arrive at it.

For a sudoku the answer is not the number of given digits, although that is what almost everything counts. There are 24 clue grids that fall to the simplest technique in the book, and 30 clue grids that will not move at all until you find an X-Wing. What decides the effort is which kinds of reasoning the grid forces on you, not how many cells arrive already filled in.

So the scale is made of techniques, ordered by what they cost the solver. Naked single and hidden single are easy; naked pair, hidden pair, pointing pair and box line reduction are medium; X-Wing and XY-Wing are hard; swordfish is expert. A logical solver applies them from the cheapest to the dearest, and the grade of the grid is the hardest one that was genuinely needed to finish it.

Two rules keep that honest. The solver goes back to the top of the scale after every successful deduction, because an X-Wing almost always unlocks a cascade of singles, and a solver that carries on down the list instead uses expensive techniques it did not need and then calls a medium grid hard. And the grade is the hardest technique used, not the last one used: a grid that needs one X-Wing and is otherwise a stroll really is hard, because a solver who does not know the X-Wing stops there, and the length of the easy part is no help at all.

There is one hard number in this area worth carrying around. The smallest number of clues a sudoku can have and still have exactly one solution is 17. Grids with 16 do not exist: McGuire, Tugemann and Civario proved it exhaustively in 2012, using roughly 7 million core hours of computing to show that no 16 clue grid has a unique answer. Anything advertising a 16 clue sudoku with a single solution is either miscounting or has more than one answer.

The clue count still matters, for a reason that is presentation and not grading. An easy grid with 24 clues does exist and does fall to singles, but it looks frightening, and whoever printed it for a grandparent throws it away before trying. So easy grids are dug down to 36 to 45 clues and expert ones to 22 to 26, and that target is applied after the grading, never in place of it.

A maze is measured from its solution, not from its size. A 25x25 maze can come out with an almost straight path, and then a big grid promises expert and plays easy. What gets measured is the share of cells lying on the solution path, from 25 per cent at easy to 55 per cent at expert, and the share of those path cells that are junctions where you can turn the wrong way, from 2 per cent at easy to 8 per cent at expert. Both are fractions of the grid, so the thresholds mean the same thing at 10x10 and at 28x28.

A word search is measured by the directions allowed and by the density. Two directions, across and down, is reading. Adding the two forward diagonals makes the search slower without changing what you do, since a diagonal still runs left to right and the eye can tell at a glance whether it sounds like a word. Adding right to left and bottom to top changes the task, because the word can no longer be read at all, only recognised backwards, and that is the real step between a pastime and a puzzle. The second lever is how tightly the grid sits on the words: 30 cells in 100 carry a word letter at easy and 58 at expert, and the empty patches that a loose grid leaves are exactly what the eye discards first.