12x12 Queens Puzzle #267
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 28-step reasoning is written out below.
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Step-by-step solution for puzzle #267
What follows is a complete solve of puzzle #267, worked the way a person works it rather than the way a computer does. It runs to 28 deductions — 12 queen placements and 16 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 shadow, region confined to one line, pairing argument on rows, pairing argument on columns, proof by contradiction, line confined to one region, last square in a column, last square in a row.
Show the full walkthrough (21 steps) — contains the solution
Step 1. Only row 8, column 8 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 8, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 2. The rose region is down to one open square, at row 12, column 11. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 12 and column 11, the remaining rose squares, and the ring of neighbours around row 12, column 11.
Step 3. The indigo region has 4 squares left, spread across rows 1 and 2, columns 1–3. 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 4. 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: 11 squares come off.
Step 5. Sea green cannot leave column 12 — every one of its open squares (3 open squares) sits there. That column belongs to sea green, so every square in it owned by another colour is dead: 5 squares in all.
The board after step 5 — 2 of 12 queens placed. Step 6. The amber region has 3 squares left, spread across rows 1 and 2, columns 9 and 10. 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 7. Wherever teal ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 8. Here is the pairing argument. The indigo and amber regions have no open squares outside rows 1 and 2 — 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. Only row 3, column 5 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 5, the rest of the blue region and every square adjacent to it — diagonals included.
Step 10. Wherever mint ends up among its 6 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. Teal is down to 3 candidates in rows 10 and 11, columns 4, 6, and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about teal itself.
The board after step 11 — 3 of 12 queens placed. Step 12. Here is the pairing argument. The amber and red regions have no open squares outside columns 9 and 10 — 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. Wherever teal ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 14. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 3. Follow the forced moves that would create and the orange region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 15. Test row 4, column 3 for mint: place a queen there, run the forced consequences through, and column 4 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 16. Wherever mint 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.
The board after step 16 — 3 of 12 queens placed. Step 17. Pale yellow is down to 3 candidates in rows 7, 9, and 10, columns 6 and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 18. The teal region is down to one open square, at row 11, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 4, the remaining teal squares, and the ring of neighbours around row 11, column 4.
Step 19. Wherever orange 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 20. Every open square left in column 6 (2 open squares) belongs to the pale yellow region. Since that column must contain a queen, pale yellow is the colour that supplies it — so pale yellow squares anywhere else on the board are out, 1 square in all.
Step 21. 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: column 7 is down to one square, putting a queen on row 4, column 7; row 5 is down to one square, putting a queen on row 5, column 1; indigo has one square left, so its queen takes row 1, column 2; amber has one square left, so its queen takes row 2, column 10; red has one square left, so its queen takes row 6, column 9; orange has one square left, so its queen takes row 9, column 3; pale yellow has one square left, so its queen takes row 7, column 6; sea green has one square left, so its queen takes row 10, column 12. 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 10, column 12.
The board after step 21 — 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 #267
12x12 puzzle #267 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.
Like every board on this site, #267 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.
How the colors divide this board
From smallest to largest, the regions on puzzle #267 are: khaki (1 square), rose (1 square), indigo (4 squares), blue (5 squares), amber (6 squares), red (8 squares), sea green (13 squares), teal (13 squares), orange (15 squares), pale yellow (20 squares), slate (23 squares), mint (35 squares). An even split would hand every color 12 squares, so the khaki region is running 11 squares under that average while the mint region is the roomiest area on the grid with 35 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Khaki has only 1 place to put its queen; mint has 35. The spread between them on this board is 34 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 khaki is the most solid, filling 100% of its 1×1 box, and pale yellow the most diffuse at 36% of its 8×7 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
Khaki: 1 square across rows 8–8, columns 8–8. It never leaves row 8, so its queen is locked to that row.
Rose covers 1 square, spanning 1 row and 1 column (rows 12–12, columns 11–11). It never leaves row 12, so its queen is locked to that row.
Indigo occupies 4 squares within rows 1–2, columns 1–3.
Blue: 5 squares across rows 1–3, columns 4–6.
Amber covers 6 squares, spanning 2 rows and 4 columns (rows 1–2, columns 9–12).
Red occupies 8 squares within rows 6–9, columns 8–11.
Sea green: 13 squares across rows 6–12, columns 10–12.
Teal covers 13 squares, spanning 3 rows and 7 columns (rows 10–12, columns 3–9).
Orange occupies 15 squares within rows 7–12, columns 1–5.
Pale yellow: 20 squares across rows 5–12, columns 5–11.
Slate covers 23 squares, spanning 9 rows and 4 columns (rows 2–10, columns 1–4).
Mint occupies 35 squares within rows 1–6, columns 2–12.
Where to start on this board
Begin where the choices are fewest. On this board that means khaki, which owns just 1 square inside a 1×1 window running from row 8 to row 8 and column 8 to column 8. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because khaki never leaves row 8, its queen must be in that row — and that instantly settles row 8 for everyone else. Sweep along it and X out every square that is not khaki. 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. Rose, 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 #267
A quick way to find the next move is to count how many different colors touch each line. On this board, row 4 is the narrowest, touched by only 2 colors. Columns 1, 7 and 12 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 8 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.
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.
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 #267 is rated medium
The medium label comes out of four measurable things about #267: four of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 34. Boards that score tight on those measures give up their first queen fast.
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.
If #267 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.
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 #267 the place to look for one is around row 4.
As a sanity check rather than a spoiler: the finished solution to #267 puts 4 queens on the outer border and 8 queens in the interior, with none in a 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 #267?
Puzzle #267 is rated medium. 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 12x12 puzzle #267 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #267, 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 #267?
Start with the khaki region — the most constrained color on this board, with 1 square spanning rows 8–8 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.