Takuzu solving techniques
From the first pair you spot to the contradiction that only Expert grids need: these are the techniques our solver uses on Takuzu, each with a real example and the share of our puzzles that needs it.
Pairstier 1
When: Two equal circles side by side: the cells at both ends take the other colour.
Two equal circles side by side are a wall: the cell before and the cell after them must be the other colour. Most grids start with a cascade of these.
From one of our puzzles (6x6): A3 and A4 are both white; a third white next to them would make three in a row, so A2 and A5 are black.
| Share of puzzles that need it | Easy (712) | Medium (712) | Hard (712) | Expert (612) |
|---|---|---|---|---|
| Pairs | 100% | 100% | 100% | 100% |
Sandwichtier 1
When: A gap between two equal circles takes the other colour.
Two equal circles with one empty cell between them: the gap must be the other colour, or the three would match.
From one of our puzzles (6x6): B5 sits between two white circles (B4 and B6), so it is black.
| Share of puzzles that need it | Easy (712) | Medium (712) | Hard (712) | Expert (612) |
|---|---|---|---|---|
| Sandwich | 100% | 100% | 100% | 100% |
Balancetier 2
When: A line that already has half of one colour is the other colour everywhere else.
A line of 10 holds five of each colour. Once a line has its five whites, every open cell left in it is black, and the other way round.
From one of our puzzles (6x6): Row 5 already has its 3 black circles, so the 2 open cells left are white.
| Share of puzzles that need it | Easy (712) | Medium (712) | Hard (712) | Expert (612) |
|---|---|---|---|---|
| Balance | 100% | 100% | 100% | 100% |
Line analysistier 3
When: List the valid ways to finish a line; cells that agree in all of them are certain.
When pairs and counting stop, list the ways the line can still be finished without breaking a rule. Cells that are the same in every way are certain. On a 10-cell line there are rarely more than a handful of ways left.
From one of our puzzles (6x6): Of the 2 valid ways to finish row 2, every one puts white in F2.
| Share of puzzles that need it | Easy (712) | Medium (712) | Hard (712) | Expert (612) |
|---|---|---|---|---|
| Line analysis | 0% | 100% | 98% | 100% |
No twin linestier 4
When: A line that could only be finished as a copy of a full line must be finished another way.
The rule people forget: no two rows (or columns) may be identical. If a line could only be finished as a copy of a full line, that finish is out, and what remains decides some cells.
From one of our puzzles (6x6): Row 2 must not copy a full row; of the 2 ways to finish it, the only one that is not a copy puts white in D2, black in E2.
| Share of puzzles that need it | Easy (712) | Medium (712) | Hard (712) | Expert (612) |
|---|---|---|---|---|
| No twin lines | 0% | 0% | 100% | 95% |
Contradictiontier 5
When: If one colour in a cell leads to a broken rule a few moves later, the cell takes the other colour.
Assume one colour in a cell and follow pairs, counting and line analysis. If a rule breaks, the cell is the other colour. Only our Expert grids need this.
From one of our puzzles (10x10): Suppose A1 were white: following the rules from there, Row 5 would have more than 5 white circles. So A1 is black.
| Share of puzzles that need it | Easy (712) | Medium (712) | Hard (712) | Expert (612) |
|---|---|---|---|---|
| Contradiction | 0% | 0% | 0% | 100% |
Source: the Gridaday bank, 2,748 puzzles of Takuzu graded by our solver; data as of 2026-10-17.