11x11 Queens Puzzle #266
Difficulty: medium. Place 11 queens on this 11x11 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 25-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 25 deductions — 11 queen placements and 14 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 proof by contradiction.
Techniques this board needs: region confined to one line, pairing argument on rows, region shadow, proof by contradiction, last square in a region, pairing argument on columns.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Slate cannot leave row 5 — every one of its open squares (2 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 13 squares in all.
Step 2. All of the orange region's remaining squares (2 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 12 squares come off.
Step 3. Orange, sea green, and teal are now boxed into rows 9–11 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 12 squares at once.
Step 4. Here is the pairing argument. The red, pale yellow, and khaki regions have no open squares outside rows 6–8 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 3 squares come off in one stroke.
Step 5. Mint is down to 4 candidates in rows 2–4, columns 10 and 11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.
The board after step 5 — 0 of 11 queens placed. Step 6. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 8. Follow the forced moves that would create and row 9 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 7. Test row 1, column 9 for indigo: place a queen there, run the forced consequences through, and the blue region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 8. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 10. Follow the forced moves that would create and row 9 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 9. Test row 1, column 11 for indigo: place a queen there, run the forced consequences through, and row 9 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 10. Wherever indigo 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 11. Test row 2, column 10 for indigo: place a queen there, run the forced consequences through, and row 9 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 11 — 0 of 11 queens placed. Step 12. The indigo region is down to one open square, at row 1, column 7. Every colour holds exactly one queen, so that square is forced. Row 1 and column 7 are now spoken for, along with the squares touching that queen.
Step 13. Mint, slate, khaki, and sea green are now boxed into columns 8–11 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 5 squares at once.
Step 14. Blue is down to 3 candidates in rows 3 and 4, columns 5 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 15. Blue and pale yellow are now boxed into columns 5 and 6 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 16. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: teal has one square left, so its queen takes row 11, column 3; orange has one square left, so its queen takes row 9, column 4; sea green has one square left, so its queen takes row 10, column 11; mint has one square left, so its queen takes row 4, column 10; blue has one square left, so its queen takes row 3, column 6; slate has one square left, so its queen takes row 5, column 8; pale yellow has one square left, so its queen takes row 7, column 5; khaki has one square left, so its queen takes row 8, column 9; red has one square left, so its queen takes row 6, column 1; amber has one square left, so its queen takes row 2, column 2. 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 2, column 2.
The board after step 16 — 11 of 11 queens placed.
That completes the board: 11 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 11x11 Queens puzzle #266
Board #266 of the 11x11 bank carries a medium rating. Its 121 squares are shared between eleven colored regions, and the solution places eleven queens: one per row, one per column, one per region, none of them side by side or corner to corner.
Like every board on this site, #266 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.
How the colors divide this board
From smallest to largest, the regions on puzzle #266 are: slate (2 squares), orange (2 squares), indigo (6 squares), sea green (6 squares), mint (9 squares), teal (12 squares), pale yellow (13 squares), blue (14 squares), khaki (16 squares), red (18 squares), amber (23 squares). An even split would hand every color 11 squares, so the slate region is running 9 squares under that average while the amber region is the roomiest area on the grid with 23 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Slate has only 2 places to put its queen; amber has 23. The spread between them on this board is 21 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Square count is only half the story — how a color is arranged counts too. Slate is the tidiest shape on #266, packing 100% of its 1×2 bounding box, whereas mint straggles across its 5×4 box at just 45%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Below is every color on #266 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Slate occupies 2 squares within rows 5–5, columns 8–9. It never leaves row 5, so its queen is locked to that row.
Orange: 2 squares across rows 9–10, columns 4–4. It sits wholly inside column 4, so its queen must be in that column.
Indigo covers 6 squares, spanning 2 rows and 5 columns (rows 1–2, columns 7–11).
Sea green occupies 6 squares within rows 9–11, columns 8–11.
Mint: 9 squares across rows 2–6, columns 8–11.
Teal covers 12 squares, spanning 3 rows and 6 columns (rows 9–11, columns 3–8).
Pale yellow occupies 13 squares within rows 5–9, columns 4–8.
Blue: 14 squares across rows 1–4, columns 5–10.
Khaki covers 16 squares, spanning 5 rows and 5 columns (rows 6–10, columns 7–11).
Red occupies 18 squares within rows 5–11, columns 1–5.
Amber: 23 squares across rows 1–7, columns 1–5.
Where to start on this board
Begin where the choices are fewest. On this board that means slate, which owns just 2 squares inside a 1×2 window running from row 5 to row 5 and column 8 to column 9. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. Then look at which colors just lost ground: on a grid this size that one observation usually forces a second region within a move or two.
Work outward to the other flat regions next. Indigo and orange are no more than two rows tall, so their queens are 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 1, 3, 4, 8 and 11 are the narrowest, touched by only 3 colors. Columns 1 and 2 are the tightest columns, with 2 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 10 rows. Row 9, meeting 6 colors, is the opposite case — the busiest line on #266 and the last one you should expect to crack.
Working through the middle game
Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.
Keep alternating between two views of the board. In the region view you ask which of the eleven colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 11x11 board unwinds.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #266 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 2-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 21. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
If #266 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.
Techniques that work well at 11x11
At 11x11 you are managing 121 squares, and holding the whole board in your head will not work. Split it into quadrants and solve locally. Most useful deductions on a grid this large are regional — a color pinched between two placed queens, or a band of rows that two regions must share. Trust the marks on the board over your memory.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #266 the place to look for one is around column 1.
Here is a shape check that gives nothing away: when #266 is solved, 4 queens sit on the outer edge of the grid and 7 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 11x11 Queens puzzle #266?
Puzzle #266 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #266 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #266, has been verified to have exactly one valid arrangement of eleven 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 slate region — the most constrained color on this board, with 2 squares spanning rows 5–5 and columns 8–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.