11x11 Queens Puzzle #41
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 23-step reasoning is written out below.
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Step-by-step solution for puzzle #41
What follows is a complete solve of puzzle #41, worked the way a person works it rather than the way a computer does. It runs to 23 deductions — 11 queen placements and 12 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: last square in a region, region shadow, region confined to one line, pairing argument on rows, proof by contradiction, last square in a row.
Show the full walkthrough (17 steps) — contains the solution
Step 1. The board opens by cascading. Each of these placements is forced the moment the one before it is marked, so the whole run is really a single sweep repeated: teal has one square left, so its queen takes row 11, column 10; indigo has one square left, so its queen takes row 1, column 9; khaki has one square left, so its queen takes row 8, column 8; orange has one square left, so its queen takes row 9, column 11. 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 9, column 11.
Step 2. Blue is down to 3 candidates in rows 2 and 3, columns 5 and 7, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about blue itself.
Step 3. The mint region has 5 squares left, spread across rows 3 and 4, columns 4–7. 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.
Step 4. Sea green cannot leave row 10 — every one of its open squares (2 open squares) sits there. That row belongs to sea green, so every square on it owned by another colour is dead: 5 squares in all.
The board after step 4 — 4 of 11 queens placed. Step 5. Pale yellow is down to 4 candidates in rows 5 and 7, columns 5–7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 6. Amber, blue, and mint are now boxed into rows 2–4 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 4 squares at once.
Step 7. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 1. Follow the forced moves that would create and row 6 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 8. Test row 2, column 3 for amber: place a queen there, run the forced consequences through, and the slate region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 9. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 4. Follow the forced moves that would create and the mint region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
The board after step 9 — 4 of 11 queens placed. Step 10. Test row 3, column 1 for amber: place a queen there, run the forced consequences through, and the red region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 11. Amber is down to 2 candidates in rows 2 and 4, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 12. Test row 4, column 1 for amber: place a queen there, run the forced consequences through, and row 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 13. The amber region is down to one open square, at row 2, column 2. Every colour holds exactly one queen, so that square is forced. Row 2 and column 2 are now spoken for, along with the squares touching that queen.
The board after step 13 — 5 of 11 queens placed. Step 14. Only row 3, column 7 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 7, the rest of the blue region and every square adjacent to it — diagonals included.
Step 15. The mint region is down to one open square, at row 4, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 4, the remaining mint squares, and the ring of neighbours around row 4, column 4.
Step 16. Wherever slate ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 17. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: row 5 is down to one square, putting a queen on row 5, column 1; sea green has one square left, so its queen takes row 10, column 5; pale yellow has one square left, so its queen takes row 7, column 6; red has one square left, so its queen takes row 6, column 3. 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 6, column 3.
The board after step 17 — 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 #41
Puzzle #41 is one of the medium-rated boards in the 11x11 set. Eleven colors divide its 121 squares, and you win by placing eleven queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
As with every puzzle in this collection, #41 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.
How the colors divide this board
From smallest to largest, the regions on puzzle #41 are: teal (1 square), indigo (2 squares), mint (7 squares), khaki (7 squares), amber (9 squares), orange (11 squares), red (12 squares), sea green (14 squares), slate (15 squares), pale yellow (18 squares), blue (25 squares). An even split would hand every color 11 squares, so the teal region is running 10 squares under that average while the blue region is the roomiest area on the grid with 25 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 teal, with 1 square, is under far more pressure than blue with 25. A 24-square gap like this one rewards players who start where the choices are fewest.
Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×2 box that encloses it, while blue is the most scattered at 45% of its 7×8 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.
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.
Teal occupies 1 square within rows 11–11, columns 10–10. It never leaves row 11, so its queen is locked to that row.
Indigo: 2 squares across rows 1–1, columns 9–10. It never leaves row 1, so its queen is locked to that row.
Mint covers 7 squares, spanning 3 rows and 4 columns (rows 2–4, columns 4–7).
Khaki occupies 7 squares within rows 8–11, columns 8–10.
Amber: 9 squares across rows 1–4, columns 1–4.
Orange covers 11 squares, spanning 7 rows and 3 columns (rows 5–11, columns 9–11).
Red occupies 12 squares within rows 6–10, columns 1–4.
Sea green: 14 squares across rows 8–11, columns 1–6.
Slate covers 15 squares, spanning 4 rows and 7 columns (rows 3–6, columns 1–7).
Pale yellow occupies 18 squares within rows 5–11, columns 5–8.
Blue: 25 squares across rows 1–7, columns 4–11.
Where to start on this board
Teal is the natural opening on #41: it is the smallest region on the grid at 1 square, boxed into rows 11–11 and columns 10–10. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.
Because teal never leaves row 11, its queen must be in that row — and that instantly settles row 11 for everyone else. Sweep along it and X out every square that is not teal. 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 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 #41
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 2 are the narrowest, touched by only 3 colors. Column 11 is the tightest column, with 2 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 10 rows. By contrast row 11 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
The middle of #41 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.
The whole middle game reduces to two sweeps. Sweep the eleven colors looking for one with a single square left; sweep the eleven rows and eleven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 11x11 solve from a stalled one.
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 #41 is rated medium
Ratings track how generous the opening is. On #41 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 24-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.
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.
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 #41, start hunting for it near column 11.
As a sanity check rather than a spoiler: the finished solution to #41 puts 4 queens on the outer border and 7 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 11x11 Queens puzzle #41?
Puzzle #41 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #41 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #41, 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 #41?
Start with the teal region — the most constrained color on this board, with 1 square spanning rows 11–11 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.