12x12 Queens Puzzle #7
Difficulty: medium. 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 #7
What follows is a complete solve of puzzle #7, 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 proof by contradiction.
Techniques this board needs: last square in a region, region confined to one line, pairing argument on rows, line confined to one region, pairing argument on columns, last square in a row, proof by contradiction.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Only row 6, column 4 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 4, the rest of the red region and every square adjacent to it — diagonals included.
Step 2. The rose region is down to one open square, at row 12, column 12. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 12 and column 12, the remaining rose squares, and the ring of neighbours around row 12, 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. All of the blue region's remaining squares (2 open squares) lie in row 3. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 10 squares come off.
Step 5. Amber cannot leave row 2 — every one of its open squares (3 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 5 squares in all.
The board after step 5 — 2 of 12 queens placed. Step 6. All of the slate 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: 7 squares come off.
Step 7. Only row 11, column 2 is still open in teal, and since each colour must take a queen, that settles it. Everything else in row 11, in column 2, in teal and in the ring around that square is now dead.
Step 8. Here is the pairing argument. The mint and slate regions have no open squares outside rows 4 and 5 — 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: 6 squares come off in one stroke.
Step 9. Only row 7, column 9 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 9, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 10. Every open square left in column 10 (2 open squares) belongs to the orange region. Since that column must contain a queen, orange is the colour that supplies it — so orange squares anywhere else on the board are out, 5 squares in all.
Step 11. Orange cannot leave column 10 — 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: 1 square in all.
The board after step 11 — 4 of 12 queens placed. Step 12. Here is the pairing argument. The indigo and blue regions have no open squares outside columns 6 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: 6 squares come off in one stroke.
Step 13. Amber and sea green are now boxed into columns 8 and 11 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 1 square at once.
Step 14. Row 5 has a single open square left, at column 1 in the slate region. Every row needs a queen, so this one is forced. Row 5 and column 1 are now spoken for, along with the squares touching that queen.
Step 15. Test row 1, column 7 for indigo: place a queen there, run the forced consequences through, and row 8 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 16. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: indigo has one square left, so its queen takes row 1, column 6; blue has one square left, so its queen takes row 3, column 7; amber has one square left, so its queen takes row 2, column 11; sea green has one square left, so its queen takes row 10, column 8; orange has one square left, so its queen takes row 9, column 10; khaki has one square left, so its queen takes row 8, column 3; mint has one square left, so its queen takes row 4, column 5. 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 5.
The board after step 16 — 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 #7
12x12 puzzle #7 is rated medium. 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 #7 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 #7 are: red (1 square), rose (1 square), indigo (2 squares), blue (4 squares), slate (4 squares), teal (9 squares), amber (12 squares), khaki (15 squares), orange (17 squares), pale yellow (19 squares), sea green (29 squares), mint (31 squares). An even split would hand every color 12 squares, so the red region is running 11 squares under that average while the mint region is the roomiest area on the grid with 31 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Red has only 1 place to put its queen; mint has 31. The spread between them on this board is 30 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 1×2 box that encloses it, while orange is the most scattered at 49% of its 5×7 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
Here is the full footprint of every color on puzzle #7, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Red covers 1 square, spanning 1 row and 1 column (rows 6–6, columns 4–4). It never leaves row 6, so its queen is locked to that row.
Rose occupies 1 square within rows 12–12, columns 12–12. It never leaves row 12, so its queen is locked to that row.
Indigo: 2 squares across rows 1–1, columns 6–7. It never leaves row 1, so its queen is locked to that row.
Blue covers 4 squares, spanning 2 rows and 2 columns (rows 2–3, columns 6–7).
Slate occupies 4 squares within rows 3–6, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Teal: 9 squares across rows 7–12, columns 1–3.
Amber covers 12 squares, spanning 3 rows and 5 columns (rows 1–3, columns 8–12).
Khaki occupies 15 squares within rows 6–10, columns 2–6.
Orange: 17 squares across rows 6–10, columns 5–11.
Pale yellow covers 19 squares, spanning 7 rows and 5 columns (rows 2–8, columns 8–12).
Sea green occupies 29 squares within rows 8–12, columns 3–12.
Mint: 31 squares across rows 1–7, columns 1–9.
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 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 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.
Work outward to the other flat regions next. Indigo, rose 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 #7
A quick way to find the next move is to count how many different colors touch each line. On this board, row 11 is the narrowest, touched by only 2 colors. Columns 1 and 2 are the tightest columns, with 3 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 11 rows. By contrast row 6 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
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.
Keep alternating between two views of the board. In the region view you ask which of the twelve colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 12x12 board unwinds.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #7 is rated medium
The rating reflects how much the board concedes up front. Puzzle #7 has six regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 30 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
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 #7 the place to look for one is around row 11.
As a sanity check rather than a spoiler: the finished solution to #7 puts 3 queens on the outer border and 9 queens in the interior, 1 of them in an actual corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 #7?
Puzzle #7 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and six of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #7 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #7, 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 #7?
Start with the red region — the most constrained color on this board, with 1 square spanning rows 6–6 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.