9x9 Queens Puzzle #252
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 16-step reasoning is written out below.
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Step-by-step solution for puzzle #252
What follows is a complete solve of puzzle #252, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 9 queen placements and 7 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: last square in a region, region shadow, region confined to one line, pairing argument on columns, line confined to one region, last square in a row.
Show the full walkthrough (9 steps) — contains the solution
Step 1. Only row 6, column 7 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 7, the rest of the red region and every square adjacent to it — diagonals included.
Step 2. Amber is down to 5 candidates in rows 1–3, columns 1–3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 3. Mint cannot leave column 1 — 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: 7 squares in all.
The board after step 3 — 1 of 9 queens placed. Step 4. Wherever amber ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Indigo is down to 4 candidates in rows 1–3, columns 4 and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 6. Here is the pairing argument. The amber and pale yellow regions have no open squares outside columns 2 and 3 — 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: 6 squares come off in one stroke.
The board after step 6 — 1 of 9 queens placed. Step 7. Wherever khaki ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 8. Every open square left in row 9 (5 open squares) belongs to the orange region. Since that row must contain a queen, orange is the colour that supplies it — so orange squares anywhere else on the board are out, 4 squares in all.
Step 9. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: row 8 is down to one square, putting a queen on row 8, column 4; indigo has one square left, so its queen takes row 1, column 5; row 7 is down to one square, putting a queen on row 7, column 2; amber has one square left, so its queen takes row 2, column 3; row 5 is down to one square, putting a queen on row 5, column 9; blue has one square left, so its queen takes row 3, column 8; mint has one square left, so its queen takes row 4, column 1; orange has one square left, so its queen takes row 9, column 6. 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 6.
The board after step 9 — 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 #252
This is puzzle number 252 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.
As with every puzzle in this collection, #252 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 #252 are: red (1 square), mint (3 squares), indigo (5 squares), amber (5 squares), khaki (10 squares), pale yellow (11 squares), orange (12 squares), slate (13 squares), blue (21 squares). An even split would hand every color 9 squares, so the red region is running 8 squares under that average while the blue region is the roomiest area on the grid with 21 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Red is working with 1 square while blue has 21 to spare, a gap of 20. 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. Mint is the tidiest shape on #252, packing 100% of its 3×1 bounding box, whereas blue straggles across its 7×7 box at just 43%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Below is every color on #252 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.
Red: 1 square across rows 6–6, columns 7–7. It never leaves row 6, so its queen is locked to that row.
Mint covers 3 squares, spanning 3 rows and 1 column (rows 2–4, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.
Indigo occupies 5 squares within rows 1–3, columns 3–5.
Amber: 5 squares across rows 1–3, columns 1–3.
Khaki covers 10 squares, spanning 4 rows and 5 columns (rows 6–9, columns 2–6).
Pale yellow occupies 11 squares within rows 3–9, columns 1–3.
Orange: 12 squares across rows 7–9, columns 4–9.
Slate covers 13 squares, spanning 5 rows and 4 columns (rows 3–7, columns 6–9).
Blue occupies 21 squares within rows 1–7, columns 3–9.
Where to start on this board
Begin where the choices are fewest. On this board that means red, which owns just 1 square inside a 1×1 window running from row 6 to row 6 and column 7 to column 7. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because red never leaves row 6, its queen must be in that row — and that instantly settles row 6 for everyone else. Sweep along it and X out every square that is not red. 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.
Nothing on this grid is pinned to one or two lines, which is why the opening takes patience. Scan the columns for one where only two or three colors appear — whichever region is forced into it claims that column outright. Chaining two of those observations is normally enough to break #252 open.
The tightest lines on puzzle #252
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 5, 8 and 9 are the narrowest, touched by only 3 colors. Columns 1, 2, 8 and 9 are the tightest columns, with 3 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 8 rows. Row 3, meeting 6 colors, is the opposite case — the busiest line on #252 and the last one you should expect to crack.
Working through the middle game
The middle of #252 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.
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.
A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.
Why puzzle #252 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers three single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 20. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.
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 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.
Here is a shape check that gives nothing away: when #252 is solved, 4 queens sit on the outer edge of the grid and 5 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 9x9 Queens puzzle #252?
Puzzle #252 is rated hard. 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 9x9 puzzle #252 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #252, 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 #252?
Start with the red region — the most constrained color on this board, with 1 square spanning rows 6–6 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.