9x9 Queens Puzzle #268
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 15-step reasoning is written out below.
Previous puzzle · All 9x9 puzzles · Next puzzle
Step-by-step solution for puzzle #268
What follows is a complete solve of puzzle #268, worked the way a person works it rather than the way a computer does. It runs to 15 deductions — 9 queen placements and 6 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 region shadow.
Techniques this board needs: last square in a region, region shadow, region confined to one line.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Only row 9, column 4 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 4, the rest of the orange region and every square adjacent to it — diagonals included.
Step 2. Indigo is down to 4 candidates in rows 1–3, columns 8 and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. Blue cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 8 squares in all.
Step 4. All of the red region's remaining squares (4 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
The board after step 4 — 1 of 9 queens placed. Step 5. 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 6. The khaki region is down to one open square, at row 8, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 7, the remaining khaki squares, and the ring of neighbours around row 8, column 7.
Step 7. Amber cannot leave row 2 — every one of its open squares (2 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 4 squares in all.
Step 8. The blue region is down to one open square, at row 3, column 1. Every colour holds exactly one queen, so that square is forced. Row 3 and column 1 are now spoken for, along with the squares touching that queen.
The board after step 8 — 4 of 9 queens placed. Step 9. Mint cannot leave column 6 — every one of its open squares (2 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 4 squares in all.
Step 10. The amber region is down to one open square, at row 2, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 5, the remaining amber squares, and the ring of neighbours around row 2, column 5.
Step 11. Slate is down to 5 candidates in rows 4–7, columns 2 and 3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.
Step 12. 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: pale yellow has one square left, so its queen takes row 7, column 3; 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 6; red has one square left, so its queen takes row 6, column 9. 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 9.
The board after step 12 — 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 #268
This is puzzle number 268 in the 9x9 collection, rated hard. The grid holds 81 squares split between nine colored regions, and a finished board carries nine queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
There is exactly one way to finish #268, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these nine regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #268 are: orange (1 square), blue (2 squares), indigo (4 squares), red (4 squares), khaki (12 squares), amber (13 squares), mint (13 squares), slate (14 squares), pale yellow (18 squares). An even split would hand every color 9 squares, so the orange region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 18 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Orange is working with 1 square while pale yellow has 18 to spare, a gap of 17. Attack the cramped end and the roomy regions tend to resolve themselves.
Square count is only half the story — how a color is arranged counts too. Blue is the tidiest shape on #268, packing 100% of its 2×1 bounding box, whereas khaki straggles across its 4×6 box at just 50%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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.
Orange covers 1 square, spanning 1 row and 1 column (rows 9–9, columns 4–4). It never leaves row 9, so its queen is locked to that row.
Blue occupies 2 squares within rows 2–3, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Indigo: 4 squares across rows 1–3, columns 8–9.
Red covers 4 squares, spanning 4 rows and 1 column (rows 4–7, columns 9–9). It sits wholly inside column 9, so its queen must be in that column.
Khaki occupies 12 squares within rows 6–9, columns 4–9.
Amber: 13 squares across rows 1–2, columns 1–8.
Mint covers 13 squares, spanning 4 rows and 6 columns (rows 2–5, columns 3–8).
Slate occupies 14 squares within rows 3–7, columns 1–5.
Pale yellow: 18 squares across rows 6–9, columns 1–7.
Where to start on this board
Begin where the choices are fewest. On this board that means orange, which owns just 1 square inside a 1×1 window running from row 9 to row 9 and column 4 to column 4. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because orange never leaves row 9, its queen must be in that row — and that instantly settles row 9 for everyone else. Sweep along it and X out every square that is not orange. 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 and blue 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 #268
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 8 are the narrowest, touched by only 2 colors. Columns 2 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 2 colors, so one of that handful must take the row and the rest are squeezed into the other 8 rows. Row 7 sits at the far end with 4 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.
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.
Resist the plausible-looking queen. On #268 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 #268 is rated hard
Ratings track how generous the opening is. On #268 you get four regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 17-square gap between the largest and smallest regions. Those four numbers largely explain the hard 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.
Stuck on #268 despite the hard rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 1 on this board.
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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 9x9 Queens puzzle #268?
Puzzle #268 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #268 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #268, 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 #268?
Start with the orange region — the most constrained color on this board, with 1 square spanning rows 9–9 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.