12x12 Queens Puzzle #270
Difficulty: hard. 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 23-step reasoning is written out below.
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Step-by-step solution for puzzle #270
What follows is a complete solve of puzzle #270, worked the way a person works it rather than the way a computer does. It runs to 23 deductions — 12 queen placements and 11 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 columns, proof by contradiction, pairing argument on rows.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Only row 1, column 7 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 7, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The slate region is down to one open square, at row 5, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 6, the remaining slate squares, and the ring of neighbours around row 5, column 6.
Step 3. The amber region has 4 squares left, spread across rows 2 and 3, 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. 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 5. Amber is down to 3 candidates in rows 2 and 3, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
The board after step 5 — 2 of 12 queens placed. Step 6. The pale yellow region has 4 squares left, spread across rows 4, 6, and 7, columns 11 and 12. 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. Rose cannot leave row 12 — every one of its open squares (2 open squares) sits there. That row belongs to rose, so every square on it owned by another colour is dead: 10 squares in all.
Step 8. 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. Row 11 and column 4 are now spoken for, along with the squares touching that queen.
Step 9. The mint region has 3 squares left, spread across rows 2–4, columns 3 and 5. 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. All of the amber 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: 5 squares come off.
Step 11. Red is down to 5 candidates in rows 4, 6, and 7, columns 2, 3, and 8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
The board after step 11 — 3 of 12 queens placed. Step 12. Here is the pairing argument. The pale yellow and rose regions have no open squares outside columns 11 and 12 — 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: 11 squares come off in one stroke.
Step 13. Test row 3, column 1 for amber: place a queen there, run the forced consequences through, and column 8 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 14. The amber region is down to one open square, at row 2, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 1, the remaining amber squares, and the ring of neighbours around row 2, column 1.
Step 15. Blue, mint, red, and pale yellow are now boxed into rows 3, 4, 6, and 7 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 7 squares at once.
Step 16. 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: khaki has one square left, so its queen takes row 8, column 9; sea green has one square left, so its queen takes row 10, column 10; 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 3; red has one square left, so its queen takes row 6, column 2; pale yellow has one square left, so its queen takes row 7, column 11; orange has one square left, so its queen takes row 9, column 5; rose has one square left, so its queen takes row 12, 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 12, column 12.
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 #270
Board #270 of the 12x12 bank carries a hard rating. Its 144 squares are shared between twelve colored regions, and the solution places twelve queens: one per row, one per column, one per region, none of them side by side or corner to corner.
As with every puzzle in this collection, #270 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 #270 are: indigo (1 square), slate (1 square), rose (2 squares), amber (6 squares), pale yellow (6 squares), teal (9 squares), khaki (10 squares), mint (12 squares), red (20 squares), sea green (21 squares), blue (28 squares), orange (28 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the orange region is the roomiest area on the grid with 28 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So indigo, with 1 square, is under far more pressure than orange with 28. A 27-square gap like this one rewards players who start where the choices are fewest.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and mint the most diffuse at 48% of its 5×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #270 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.
Indigo: 1 square across rows 1–1, columns 7–7. It never leaves row 1, so its queen is locked to that row.
Slate covers 1 square, spanning 1 row and 1 column (rows 5–5, columns 6–6). It never leaves row 5, so its queen is locked to that row.
Rose occupies 2 squares within rows 12–12, columns 11–12. It never leaves row 12, so its queen is locked to that row.
Amber: 6 squares across rows 1–3, columns 1–3.
Pale yellow covers 6 squares, spanning 4 rows and 2 columns (rows 4–7, columns 11–12).
Teal occupies 9 squares within rows 10–12, columns 2–7.
Khaki: 10 squares across rows 4–8, columns 7–10.
Mint covers 12 squares, spanning 5 rows and 5 columns (rows 1–5, columns 2–6).
Red occupies 20 squares within rows 4–8, columns 1–8.
Sea green: 21 squares across rows 7–12, columns 8–12.
Blue covers 28 squares, spanning 6 rows and 8 columns (rows 1–6, columns 5–12).
Orange occupies 28 squares within rows 7–12, columns 1–8.
Where to start on this board
Indigo is the natural opening on #270: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 7–7. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Slate and rose 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 #270
A quick way to find the next move is to count how many different colors touch each line. On this board, row 9 is the narrowest, touched by only 2 colors. Columns 1, 9 and 10 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 2 colors, so one of that handful must take the row and the rest are squeezed into the other 11 rows. Row 5 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.
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.
The whole middle game reduces to two sweeps. Sweep the twelve colors looking for one with a single square left; sweep the twelve rows and twelve columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 12x12 solve from a stalled one.
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 #270 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers five single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 27. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
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 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.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #270, start hunting for it near row 9.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 9 queens in the middle, including 1 in a corner. 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 #270?
Puzzle #270 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #270 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #270, 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 #270?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.