12x12 Queens Puzzle #131
Difficulty: easy. Place 12 queens on this 12x12 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 22-step reasoning is written out below.
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Step-by-step solution for puzzle #131
What follows is a complete solve of puzzle #131, worked the way a person works it rather than the way a computer does. It runs to 22 deductions — 12 queen placements and 10 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 confined to one line, region shadow, pairing argument on rows, pairing argument on columns.
Show the full walkthrough (15 steps) — contains the solution
Step 1. Only row 12, column 11 is still open in rose, and since each colour must take a queen, that settles it. That clears row 12, column 11, the rest of the rose region and every square adjacent to it — diagonals included.
Step 2. The sea green region is down to one open square, at row 10, column 12. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 12, the remaining sea green squares, and the ring of neighbours around row 10, column 12.
Step 3. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 10 squares in all.
Step 4. 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. Row 2 and column 6 are now spoken for, along with the squares touching that queen.
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.
The board after step 5 — 4 of 12 queens placed. Step 6. All of the teal region's remaining squares (2 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 6 squares come off.
Step 7. Wherever red 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.
Step 8. Here is the pairing argument. The orange and teal regions have no open squares outside rows 9 and 11 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 9 squares come off in one stroke.
Step 9. The pale yellow region has 5 squares left, spread across rows 7 and 8, columns 2–5. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 10. All of the red region's remaining squares (2 open squares) lie in row 6. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.
The board after step 10 — 4 of 12 queens placed. Step 11. Khaki cannot leave row 8 — every one of its open squares (2 open squares) sits there. That row belongs to khaki, so every square on it owned by another colour is dead: 4 squares in all.
Step 12. All of the pale yellow region's remaining squares (2 open squares) lie in row 7. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 3 squares come off.
Step 13. Wherever slate ends up among its 2 squares, the same squares fall within reach of it — 4 squares in total, impossible regardless of how the colour resolves.
Step 14. Here is the pairing argument. The slate, khaki, and orange regions have no open squares outside columns 7, 9, and 10 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 2 squares come off in one stroke.
Step 15. 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: mint has one square left, so its queen takes row 4, column 5; pale yellow has one square left, so its queen takes row 7, column 4; red has one square left, so its queen takes row 6, column 2; blue has one square left, so its queen takes row 3, column 3; slate has one square left, so its queen takes row 5, column 9; khaki has one square left, so its queen takes row 8, column 7; orange has one square left, so its queen takes row 9, column 10; teal has one square left, so its queen takes row 11, column 1. 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 11, column 1.
The board after step 15 — 12 of 12 queens placed.
That completes the board: 12 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 12x12 Queens puzzle #131
12x12 puzzle #131 is rated easy. Twelve colored regions cover its 144 squares, and your job is to seat twelve queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Puzzle #131 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 #131 are: rose (1 square), indigo (2 squares), amber (4 squares), teal (6 squares), sea green (7 squares), red (8 squares), pale yellow (8 squares), orange (15 squares), mint (18 squares), blue (21 squares), slate (25 squares), khaki (29 squares). An even split would hand every color 12 squares, so the rose region is running 11 squares under that average while the khaki 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. Rose has only 1 place to put its queen; khaki 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.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×2 box, and orange the most diffuse at 42% of its 6×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #131 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.
Rose occupies 1 square within rows 12–12, columns 11–11. It never leaves row 12, so its queen is locked to that row.
Indigo: 2 squares across rows 1–1, columns 7–8. It never leaves row 1, so its queen is locked to that row.
Amber covers 4 squares, spanning 2 rows and 3 columns (rows 1–2, columns 6–8).
Teal occupies 6 squares within rows 9–12, columns 1–3.
Sea green: 7 squares across rows 10–12, columns 10–12.
Red covers 8 squares, spanning 5 rows and 3 columns (rows 4–8, columns 1–3).
Pale yellow occupies 8 squares within rows 7–9, columns 2–5.
Orange: 15 squares across rows 7–12, columns 7–12.
Mint covers 18 squares, spanning 6 rows and 6 columns (rows 1–6, columns 5–10).
Blue occupies 21 squares within rows 1–5, columns 1–5.
Slate: 25 squares across rows 1–8, columns 7–12.
Khaki covers 29 squares, spanning 7 rows and 8 columns (rows 6–12, columns 2–9).
Where to start on this board
Begin where the choices are fewest. On this board that means rose, which owns just 1 square inside a 1×1 window running from row 12 to row 12 and column 11 to column 11. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because rose never leaves row 12, its queen must be in that row — and that instantly settles row 12 for everyone else. Sweep along it and X out every square that is not rose. 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 and amber 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 #131
A quick way to find the next move is to count how many different colors touch each line. On this board, row 3 is the narrowest, touched by only 3 colors. Columns 1, 4, 6 and 12 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 11 rows. Row 12 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
The middle of #131 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 twelve queens to seat and twelve regions competing over 144 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 12x12 boards quickly and players who stall halfway.
Resist the plausible-looking queen. On #131 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 #131 is rated easy
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 28. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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 12x12
At 12x12 you are managing 144 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.
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 #131 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 8 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 12x12 Queens puzzle #131?
Puzzle #131 is rated easy. 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 12x12 puzzle #131 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #131, has been verified to have exactly one valid arrangement of twelve queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #131?
Start with the rose region — the most constrained color on this board, with 1 square spanning rows 12–12 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.