9x9 Queens Puzzle #31
Difficulty: medium. 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 #31
What follows is a complete solve of puzzle #31, 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 rows.
Techniques this board needs: region shadow, last square in a region, region confined to one line, pairing argument on rows.
Show the full walkthrough (10 steps) — contains the solution
Step 1. Wherever indigo ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 2. 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: amber has one square left, so its queen takes row 2, column 3; indigo has one square left, so its queen takes row 1, column 1; blue has one square left, so its queen takes row 3, column 9; slate has one square left, so its queen takes row 5, 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 5, column 2.
Step 3. All of the pale yellow 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: 5 squares come off.
The board after step 3 — 4 of 9 queens placed. Step 4. Orange cannot leave row 9 — every one of its open squares (2 open squares) sits there. That row belongs to orange, so every square on it owned by another colour is dead: 3 squares in all.
Step 5. Khaki is down to 3 candidates in rows 7 and 8, columns 7 and 8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 6. Pale yellow and khaki are now boxed into rows 7 and 8 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 1 square at once.
Step 7. All of the mint region's remaining squares (2 open squares) lie in column 5. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
The board after step 7 — 4 of 9 queens placed. Step 8. Only row 9, column 6 is still open in orange, and since each colour must take a queen, that settles it. Everything else in row 9, in column 6, in orange and in the ring around that square is now dead.
Step 9. The red region has 3 squares left, spread across rows 4 and 6, columns 7 and 8. 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 10. 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: khaki has one square left, so its queen takes row 8, column 8; pale yellow has one square left, so its queen takes row 7, column 4; mint has one square left, so its queen takes row 4, column 5; red has one square left, so its queen takes row 6, 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 6, column 7.
The board after step 10 — 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 #31
9x9 puzzle #31 is rated medium. 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.
Puzzle #31 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #31 are: amber (2 squares), indigo (3 squares), orange (4 squares), blue (5 squares), khaki (8 squares), slate (10 squares), pale yellow (12 squares), mint (15 squares), red (22 squares). An even split would hand every color 9 squares, so the amber region is running 7 squares under that average while the red region is the roomiest area on the grid with 22 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 2 squares while red has 22 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. Amber is the tidiest shape on #31, packing 100% of its 2×1 bounding box, whereas mint straggles across its 7×5 box at just 43%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a medium board.
Amber covers 2 squares, spanning 2 rows and 1 column (rows 1–2, columns 3–3). It sits wholly inside column 3, so its queen must be in that column.
Indigo occupies 3 squares within rows 1–2, columns 1–2.
Orange: 4 squares across rows 8–9, columns 4–6.
Blue covers 5 squares, spanning 3 rows and 2 columns (rows 1–3, columns 8–9).
Khaki occupies 8 squares within rows 7–9, columns 7–9.
Slate: 10 squares across rows 2–7, columns 1–3.
Pale yellow covers 12 squares, spanning 4 rows and 4 columns (rows 6–9, columns 1–4).
Mint occupies 15 squares within rows 2–8, columns 2–6.
Red: 22 squares across rows 1–7, columns 4–9.
Where to start on this board
Amber is the natural opening on #31: it is the smallest region on the grid at 2 squares, boxed into rows 1–2 and columns 3–3. Since one of those squares must hold a queen, that block of 2 rows and 1 column is the most contested part of the board.
Because amber stays inside column 3, its queen belongs to that column and no other region may use it. Cross out the rest of column 3 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
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 #31
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 4, 5 and 9 are the narrowest, touched by only 3 colors. Column 7 is the tightest column, with 2 colors.
A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 8 rows. Row 2 is the crowded extreme at 6 colors — worth revisiting later, not now.
Working through the middle game
The middle of #31 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.
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.
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 #31 is rated medium
The medium label comes out of four measurable things about #31: one of its regions is locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 20. Boards that score tight on those measures give up their first queen fast.
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.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
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 column 7 on this board.
A non-spoiling distribution check: solved, #31 carries 3 queens on the perimeter against 6 queens inside, 1 of them cornered. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.
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 #31?
Puzzle #31 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #31 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #31, 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 #31?
Start with the amber region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.