THE THREE RULES — QUICK REFERENCE
- No three consecutive: no row or column can have three (or more) of the same digit in a row.
- Balance: each row and column has exactly as many 0s as 1s (equal halves).
- Unique rows and columns: no two rows can be identical, no two columns can be identical.
Rule 1 creates the most forced placements. Rule 2 clears cells once one digit fills its quota. Rule 3 breaks symmetry near the end. Apply them in this order on every pass.
TECHNIQUE 1 — THREE-IN-A-ROW FORCING
Two consecutive identical digits force the cells on both sides:
_ 1 1 _ → 0 1 1 0 (both ends forced)
1 _ 1 → 1 0 1 (gap forced)
Scan every row and column on your first pass looking for these patterns. They're the fastest source of certain placements — no counting or arithmetic required.
The two-with-a-gap pattern (1 _ 1) is easy to miss. Read the row as a whole, not cell by cell.
TECHNIQUE 2 — BALANCE ARITHMETIC
In an 8×8 grid, every row and column must contain exactly four 0s and four 1s. Count the placed digits in each row and column.
When a row already has four 0s, every remaining empty cell in that row is a 1. When a row has four 1s, every remaining cell is a 0.
In a 6×6 grid: three of each. In a 10×10: five of each. The arithmetic scales with grid size — always half and half.
Balance arithmetic often clears the last 2–3 cells of a row or column in one pass, especially mid-solve when several cells are already placed.
TECHNIQUE 3 — UNIQUENESS ELIMINATION
If two rows are identical except for their empty cells, those empty cells must differ — otherwise the rows would become identical, which violates rule 3.
Example: Row A is 1 0 1 0 _ _ and Row B is 1 0 1 0 _ _ with the same filled pattern. The two empty cells at positions 5 and 6 can't both be the same in both rows. If you can figure out one of them from another technique, the other is forced.
This technique is most useful in the final quarter of the board when most cells are filled. Earlier in the solve, row/column patterns aren't distinct enough for it to help.
THE SOLVING ORDER
- Scan all rows and columns for two-consecutive or two-with-gap patterns (Rule 1). Place all forced cells.
- Count placed digits per row and column. If any row/column has reached its quota, fill the remainder (Rule 2).
- Repeat steps 1–2 until no new cells appear.
- Look for near-identical rows or columns and apply uniqueness elimination (Rule 3).
- Go back to step 1 — new placements usually unlock more forcing.
Steps 1–2 alone solve almost every easy and most medium Takuzu grids. Rule 3 is the tiebreaker when the board stalls.
THE BEGINNER TRAP
Applying the rules to individual cells instead of sweeping the whole board. "I see a two-consecutive here, place one more" — then stopping. Each rule pass should cover every row and every column before you move to the next technique. A placement in column 3 often unlocks a deduction in row 7 that you'd miss if you're focused on a single region.
Takuzu always has exactly one solution and it's always reachable without guessing. If you feel stuck, a forcing pattern or a balance count somewhere hasn't been applied yet.