9x9 Queens Puzzle #291
Difficulty: easy. 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 18-step reasoning is written out below.
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Step-by-step solution for puzzle #291
What follows is a complete solve of puzzle #291, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 9 queen placements and 9 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, pairing argument on columns.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Only row 1, column 8 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 8, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The amber region is down to one open square, at row 2, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 6, the remaining amber squares, and the ring of neighbours around row 2, column 6.
Step 3. Only row 3, column 9 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 9, in blue and in the ring around that square is now dead.
Step 4. Wherever khaki ends up among its 3 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
The board after step 4 — 3 of 9 queens placed. Step 5. Orange cannot leave column 5 — every one of its open squares (2 open squares) sits there. That column belongs to orange, so every square in it owned by another colour is dead: 5 squares in all.
Step 6. Here is the pairing argument. The red, pale yellow, khaki, and orange regions have no open squares outside rows 6–9 — 4 colours confined to 4 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 7. Test row 4, column 2 for mint: 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 8. Here is the pairing argument. The mint and red regions have no open squares outside columns 4 and 7 — 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: 1 square come off in one stroke.
Step 9. Test row 4, column 4 for mint: place a queen there, run the forced consequences through, and row 7 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 9 — 3 of 9 queens placed. Step 10. All of the mint region's remaining squares (2 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 11. Red cannot leave column 4 — every one of its open squares (2 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 2 squares in all.
Step 12. The pale yellow region has 5 squares left, spread across rows 6–9, columns 1 and 2. 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 13. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: khaki has one square left, so its queen takes row 8, column 3; red has one square left, so its queen takes row 6, column 4; orange has one square left, so its queen takes row 9, column 5; pale yellow has one square left, so its queen takes row 7, column 1; slate has one square left, so its queen takes row 5, column 2; mint has one square left, so its queen takes row 4, column 7. 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 4, column 7.
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 #291
9x9 puzzle #291 is rated easy. Nine colored regions cover its 81 squares, and your job is to seat nine queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Like every board on this site, #291 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 #291 are: indigo (1 square), khaki (3 squares), orange (3 squares), blue (6 squares), amber (7 squares), pale yellow (7 squares), slate (9 squares), red (16 squares), mint (29 squares). An even split would hand every color 9 squares, so the indigo region is running 8 squares under that average while the mint region is the roomiest area on the grid with 29 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 1 place to put its queen; mint has 29. The spread between them on this board is 28 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×1 box that encloses it, while mint is the most scattered at 40% of its 8×9 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.
Indigo: 1 square across rows 1–1, columns 8–8. It never leaves row 1, so its queen is locked to that row.
Khaki covers 3 squares, spanning 2 rows and 2 columns (rows 7–8, columns 2–3).
Orange occupies 3 squares within rows 8–9, columns 5–6.
Blue: 6 squares across rows 1–4, columns 8–9.
Amber covers 7 squares, spanning 2 rows and 5 columns (rows 1–2, columns 3–7).
Pale yellow occupies 7 squares within rows 6–9, columns 1–3.
Slate: 9 squares across rows 2–6, columns 1–4.
Red covers 16 squares, spanning 5 rows and 7 columns (rows 5–9, columns 3–9).
Mint occupies 29 squares within rows 1–8, columns 1–9.
Where to start on this board
Indigo is the natural opening on #291: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 8–8. 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 indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Amber, khaki 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 #291
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3, 4, 5 and 9 are the narrowest, touched by only 3 colors. Columns 1, 7 and 9 are the tightest columns, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 8 rows. Row 8 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
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.
With nine queens to seat and nine regions competing over 81 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 9x9 boards quickly and players who stall halfway.
Resist the plausible-looking queen. On #291 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #291 is rated easy
Ratings track how generous the opening is. On #291 you get two regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 28-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.
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 #291 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 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.
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 #291 the place to look for one is around row 3.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 5 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #291?
Puzzle #291 is rated easy. Its smallest color region holds 1 square, 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 9x9 puzzle #291 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #291, 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 #291?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.