Puzzle statistics
Every board in the library has been run through the same logical solver that produces the step-by-step walkthroughs on each puzzle page. That solver only uses named human techniques — region and line sweeps, line locks, region shadows, the pairing argument and proof by contradiction — and never guesses. The numbers below are what came back.
This is measured data about this collection of puzzles, not a summary of anyone else's. It is regenerated whenever the banks change.
Verification: no board needs a guess
1,800 of 1,800 puzzles were solved end to end by deduction alone. Where a site claims its puzzles need no guessing, this is what that ought to mean: each board was actually solved without one, and the sequence of deductions is published on the puzzle's own page. Last measured 2026-09-23.
How long a solve runs, by grid size
A step is one deduction: either a queen forced onto a square, or a rule that removes candidate squares. Larger grids need more of them, but not proportionally more — a 12x12 board has nearly three times the squares of a 7x7 and takes under twice as many deductions.
| Grid | Puzzles | Shortest | Median | Mean | Longest | Solved by logic |
|---|---|---|---|---|---|---|
| 7x7 | 300 | 7 | 11 | 11.2 | 21 | 300/300 |
| 8x8 | 300 | 8 | 13 | 13 | 25 | 300/300 |
| 9x9 | 300 | 9 | 14 | 14.7 | 29 | 300/300 |
| 10x10 | 300 | 10 | 17 | 16.7 | 32 | 300/300 |
| 11x11 | 300 | 11 | 18 | 18.9 | 66 | 300/300 |
| 12x12 | 300 | 12 | 20 | 20.9 | 56 | 300/300 |
How often each technique is needed
Share of boards at each size on which the technique is required at least once. The two sweeps appear almost everywhere; the pairing argument and proof by contradiction become steadily more necessary as grids grow, which is a fair description of what makes a large board hard.
| Technique | 7x7 | 8x8 | 9x9 | 10x10 | 11x11 | 12x12 |
|---|---|---|---|---|---|---|
| last square in a region | 100% | 100% | 100% | 100% | 100% | 100% |
| last square in a row | 18% | 20% | 25% | 32% | 32% | 35% |
| last square in a column | 19% | 18% | 18% | 25% | 23% | 26% |
| region confined to one line | 86% | 86% | 92% | 94% | 96% | 96% |
| region shadow | 74% | 81% | 84% | 90% | 92% | 90% |
| line confined to one region | 14% | 19% | 20% | 26% | 29% | 27% |
| pairing argument on rows | 14% | 23% | 29% | 41% | 52% | 54% |
| pairing argument on columns | 12% | 17% | 24% | 28% | 36% | 42% |
| proof by contradiction | 8% | 8% | 10% | 10% | 16% | 23% |
Board shape
Two structural properties predict most of a board's difficulty. The first is whether any colour is confined to a single row or column, which hands you a free elimination sweep before you place anything. The second is how small the smallest colour is, since that is where a solve should begin.
| Grid | Boards with a single-line colour | Mean smallest colour | Mean largest colour | Queens on the border |
|---|---|---|---|---|
| 7x7 | 97% | 1.4 squares | 15.6 squares | 50% |
| 8x8 | 95% | 1.4 squares | 18.7 squares | 44% |
| 9x9 | 98% | 1.3 squares | 21.5 squares | 40% |
| 10x10 | 96% | 1.3 squares | 25.2 squares | 36% |
| 11x11 | 96% | 1.3 squares | 27.7 squares | 33% |
| 12x12 | 97% | 1.2 squares | 31.3 squares | 30% |