Every puzzle generated on this site is checked for a unique solution and a logical path that does not require guessing. The key is knowing which pattern to look for next. Four strategies cover most of the forced moves in these puzzles, from a simple 6×6 to a challenging 10×10 grid. Work through them in order to find the next step.
The no-three-in-a-row rule is the most productive source of forced moves. Whenever two identical digits appear side by side, placing a third identical digit on either end would immediately create an illegal triplet. Therefore, both neighbors of the pair must be the opposite digit.
This works in every direction. Scan every row and every column for adjacent pairs at the start of each solving pass. On harder puzzles, one forced cell from the adjacent pair rule creates a new adjacent pair in the same column, which forces another cell, which cascades further. Always re-scan after each fill.
This strategy alone solves most easy puzzles completely and fills 30–50% of cells in medium and hard puzzles before you need to reach for the next technique.
A variation of the adjacent pair rule. When two identical digits are separated by exactly one empty cell — a sandwich pattern — filling the middle with the same digit would create three identical digits in a row. Therefore, the middle cell must be the opposite digit.
The sandwich pattern is easy to miss when scanning quickly. Look for any pair of identical digits with a single empty cell between them, including patterns that span across already-filled cells. Columns are just as likely to contain this pattern as rows.
Together, adjacent pair and sandwich apply the no-three-in-a-row rule exhaustively. Whenever you are stuck, double-check that you have applied both to every row and column before moving to the balance-counting strategy.
Every row and every column must contain exactly half 0s and half 1s. In an 8×8 grid, that is four 0s and four 1s per line. In a 6×6 grid, three of each. When a row already contains its full quota of one digit, every remaining empty cell in that row must be the other digit.
This is the most explosive strategy when it fires: a single row observation fills all remaining cells in that row at once. Count every partially-filled row and column. Even if a row is only two-thirds complete, knowing it has three 1s remaining can tell you that the next two cells you fill must be 0s to avoid overshooting the quota.
Apply balance counting after the adjacent pair and sandwich passes. Each forced cell from those strategies changes the count, and a row that was not yet triggerable may become triggerable after a single new fill.
No two rows may be identical, and no two columns may be identical. This rule becomes actionable when a nearly-complete row has only one empty cell remaining and it looks identical to a fully-completed row in every other position. The empty cell must be filled with the opposite value of what the completed row has in that position.
Check this strategy whenever a row or column reaches its last empty cell. Compare that row against every fully-completed row. If they match everywhere except the empty cell, the empty cell is forced. The same logic applies symmetrically to columns.
On 10×10 grids, duplicate prevention often triggers multiple times in the endgame, when most rows are nearly complete and many completed rows exist to compare against. This strategy, combined with balance counting, typically finishes the final third of a hard puzzle.
Apply all four strategies in a repeating loop. One pass rarely exhausts all forced moves; each newly filled cell opens up new deductions in the same row, column, and across the grid.
| Order | Strategy | When to Apply |
|---|---|---|
| 1 | Adjacent Pair | Always first — scan every row and column for two identical digits in a row. |
| 2 | Sandwich | After adjacent pair — scan for identical digits with one gap between them. |
| 3 | Balance Counting | Check if any row or column has hit its limit for one digit; fill the rest. |
| 4 | Duplicate Prevention | Check near-complete rows and columns against completed ones for forced differences. |
Repeat the loop until the puzzle is solved. Most easy puzzles need only one or two full passes. Hard 10×10 puzzles may require five or six. The puzzles generated here are checked so that repeating these logical steps can reveal the solution without guessing.
Start with the adjacent pair rule: any two identical digits sitting next to each other force both neighbors to be the opposite value. This creates a chain reaction that quickly fills many cells. Apply this rule to every row and column before trying anything else.
Every row and column must have exactly equal numbers of 0s and 1s. In an 8×8 grid, that means four 0s and four 1s per row. Count the filled cells. If a row already has four 1s, every remaining empty cell must be 0. This often fills two or three cells at once.
The sandwich rule applies when two identical digits have exactly one empty cell between them: for example 1, ?, 1. The middle cell must be the opposite digit (0), because placing a 1 there would create three consecutive 1s, violating the rules.
No two rows or columns may be identical. If a nearly-complete row has only one empty cell and matches a completed row in every other position, the empty cell must be the opposite of what the completed row has there. This forces a value without any counting.
Work in this order: (1) Adjacent pair, (2) Sandwich, (3) Balance counting, (4) Duplicate prevention. Repeat until solved. Most puzzles need only two or three full passes.
The puzzles generated here are checked for a logical path that does not require guessing. The four strategies above are enough for this site’s puzzles; other Takuzu collections may use additional techniques. If you feel stuck here, re-scan every row and column from the beginning.
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