9x9 Queens Puzzle #266
Difficulty: hard. Place 9 queens on this 9x9 board so that no two share a row, column or color region and none touch — including diagonally. This puzzle has exactly one solution, and it has been verified solvable by deduction alone: the complete 19-step reasoning is written out below.
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Step-by-step solution for puzzle #266
What follows is a complete solve of puzzle #266, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 9 queen placements and 10 elimination steps that narrow the board between them. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the pairing argument on columns.
Techniques this board needs: pairing argument on rows, region confined to one line, last square in a row, pairing argument on columns, region shadow, last square in a region.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Indigo, amber, blue, and mint are now boxed into rows 1–4 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 9 squares at once.
Step 2. All of the slate region's remaining squares (2 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 10 squares come off.
Step 3. Look along row 4: one square survives, row 4, column 9, which belongs to mint. A row cannot be left empty, so the queen goes there. That clears row 4, column 9, the rest of the mint region and every square adjacent to it — diagonals included.
Step 4. Here is the pairing argument. The red and pale yellow regions have no open squares outside rows 6 and 7 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 4 squares come off in one stroke.
The board after step 4 — 1 of 9 queens placed. Step 5. Blue and slate are now boxed into columns 1 and 2 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 6 squares at once.
Step 6. The pale yellow region has 3 squares left, spread across rows 6 and 7, columns 3 and 4. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 7. Wherever orange ends up among its 2 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
Step 8. Pale yellow is down to 2 candidates in rows 6 and 7, columns 3 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 9. The red region has 3 squares left, spread across rows 6 and 7, columns 5 and 6. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.
The board after step 9 — 1 of 9 queens placed. Step 10. Here is the pairing argument. The pale yellow and orange regions have no open squares outside columns 3 and 4 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 5 squares come off in one stroke.
Step 11. Only row 1, column 5 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 5, in indigo and in the ring around that square is now dead.
Step 12. All of the red region's remaining squares (2 open squares) lie in column 6. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 13. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 7; blue has one square left, so its queen takes row 3, column 1; slate has one square left, so its queen takes row 5, column 2; pale yellow has one square left, so its queen takes row 7, column 4; red has one square left, so its queen takes row 6, column 6; orange has one square left, so its queen takes row 9, column 3; khaki has one square left, so its queen takes row 8, column 8. After each one, clear that row, that column, the rest of the colour and the ring of neighbours before looking for the next. The last of them seats the queen on row 8, column 8.
The board after step 13 — 9 of 9 queens placed.
That completes the board: 9 queens seated, one in every row, one in every column and one in every colour, with no two of them touching. Notice that nothing in the sequence above required a guess — each placement followed from an elimination already on the grid, which is what “single solution, solvable by logic” actually means in practice.
About 9x9 Queens puzzle #266
Board #266 of the 9x9 bank carries a hard rating. Its 81 squares are shared between nine colored regions, and the solution places nine queens: one per row, one per column, one per region, none of them side by side or corner to corner.
Puzzle #266 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #266 are: indigo (6 squares), blue (6 squares), mint (6 squares), slate (6 squares), orange (7 squares), pale yellow (8 squares), amber (9 squares), red (15 squares), khaki (18 squares). An even split would hand every color 9 squares, so the indigo region is running 3 squares under that average while the khaki region is the roomiest area on the grid with 18 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So indigo, with 6 squares, is under far more pressure than khaki with 18. A 12-square gap like this one rewards players who start where the choices are fewest.
A region's outline can constrain it more than its size does. Here orange is the most solid, filling 88% of its 2×4 box, and khaki the most diffuse at 60% of its 5×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Indigo occupies 6 squares within rows 1–3, columns 3–5.
Blue: 6 squares across rows 1–4, columns 1–2.
Mint covers 6 squares, spanning 3 rows and 3 columns (rows 2–4, columns 7–9).
Slate occupies 6 squares within rows 3–5, columns 1–3.
Orange: 7 squares across rows 8–9, columns 1–4.
Pale yellow covers 8 squares, spanning 3 rows and 4 columns (rows 5–7, columns 1–4).
Amber occupies 9 squares within rows 1–3, columns 5–9.
Red: 15 squares across rows 4–7, columns 4–8.
Khaki covers 18 squares, spanning 5 rows and 6 columns (rows 5–9, columns 4–9).
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 6 squares between rows 1 and 3 and columns 3 and 5. Its queen has nowhere else to go, which means those 3 rows and 3 columns carry more constraint than anywhere else on #266.
Indigo does not collapse onto a single line, so it will not hand you a queen on move one. What it does give you is a short list of 6 candidates. Take each one in turn, imagine the queen there, cross out its row, its column and the eight squares around it, and ask whether any other region is left with nowhere to go. One or two candidates usually fail that test immediately.
Work outward to the other flat regions next. Orange is no more than two rows tall, so its queen is confined to a narrow band. When two flat regions overlap the same pair of rows, those rows are spoken for and every other color is locked out of both — the single most productive pattern in the game, and it is live on this board.
The tightest lines on puzzle #266
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 8 and 9 are the narrowest, touched by only 2 colors. Columns 6 and 9 are the tightest columns, with 3 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 8 rows. By contrast row 3 meets 5 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
Two checks, run alternately, will carry you through the 81 squares of this grid. First: has any of the nine regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 9x9 boards almost always run one of those checks and forget the other.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #266 is rated hard
Ratings track how generous the opening is. On #266 you get no region pinned to a single row or column, a smallest color of 6 squares, a tightest row crossed by 2 colors, and a 12-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a hard board from the rest.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
Techniques that work well at 9x9
At 9x9 a full re-scan after every move starts to get slow, so work locally instead. Once a queen goes down, check only the regions, rows and columns that placement actually touched — that is where new information appeared. Fall back to a complete sweep only when the local pass turns up nothing, and your solve times will drop noticeably.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #266, start hunting for it near row 8.
As a sanity check rather than a spoiler: the finished solution to #266 puts 4 queens on the outer border and 5 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
How to play Queens
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- Tap once to mark a square with an X (where a queen can't go), tap again to place a queen, and once more to clear it.
- There is always exactly one valid solution. Use logic, never guessing, to find it.
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Questions about this puzzle
How hard is 9x9 Queens puzzle #266?
Puzzle #266 is rated hard. Its smallest color region holds 6 squares, its tightest row is touched by 2 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 9x9 puzzle #266 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #266, has been verified to have exactly one valid arrangement of nine queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #266?
Start with the indigo region — the most constrained color on this board, with 6 squares spanning rows 1–3 and columns 3–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.