12x12 Queens Puzzle #262
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 31-step reasoning is written out below.
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Step-by-step solution for puzzle #262
What follows is a complete solve of puzzle #262, worked the way a person works it rather than the way a computer does. It runs to 31 deductions — 12 queen placements and 19 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, last square in a column, line confined to one region, last square in a row.
Show the full walkthrough (23 steps) — contains the solution
Step 1. Only row 1, column 12 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 12, the rest of the indigo 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 amber region has 6 squares left, spread across rows 2 and 3, columns 2–6. 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 red ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 5. Sea green cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to sea green, so every square in it owned by another colour is dead: 10 squares in all.
Step 6. All of the pale yellow region's remaining squares (3 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
The board after step 6 — 2 of 12 queens placed. Step 7. Wherever amber 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. Red is down to 3 candidates in rows 5 and 6, columns 3 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 9. Amber, mint, red, and orange are now boxed into columns 3–6 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 16 squares at once.
Step 10. All of the teal region's remaining squares (2 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 9 squares come off.
Step 11. Blue is down to 6 candidates in rows 2–4, columns 8–10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 12. Now a short proof by contradiction. Suppose the amber queen went to row 2, 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.
The board after step 12 — 2 of 12 queens placed. Step 13. Test row 2, column 4 for amber: place a queen there, run the forced consequences through, and column 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 14. Amber is down to 3 candidates in rows 2 and 3, columns 5 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 15. Test row 3, column 5 for amber: place a queen there, run the forced consequences through, and column 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 16. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 9. Follow the forced moves that would create and column 6 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 17. Test row 4, column 10 for blue: place a queen there, run the forced consequences through, and column 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 17 — 2 of 12 queens placed. Step 18. Here is the pairing argument. The amber and blue regions have no open squares outside rows 2 and 3 — 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: 3 squares come off in one stroke.
Step 19. Test row 4, column 3 for mint: place a queen there, run the forced consequences through, and the orange region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 20. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 6. 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 21. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 5; blue has one square left, so its queen takes row 3, column 10; column 6 is down to one square, putting a queen on row 4, column 6; pale yellow has one square left, so its queen takes row 7, column 2; column 4 is down to one square, putting a queen on row 6, column 4. 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 4.
Step 22. Every open square left in row 5 (2 open squares) belongs to the slate region. Since that row must contain a queen, slate is the colour that supplies it — so slate squares anywhere else on the board are out, 1 square in all.
Step 23. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: row 8 is down to one square, putting a queen on row 8, column 8; slate has one square left, so its queen takes row 5, column 9; teal has one square left, so its queen takes row 11, column 7; sea green has one square left, so its queen takes row 10, column 1; orange has one square left, so its queen takes row 9, column 3. 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 9, column 3.
The board after step 23 — 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 #262
Board #262 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, #262 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 #262 are: indigo (1 square), sea green (3 squares), rose (3 squares), red (4 squares), mint (9 squares), pale yellow (10 squares), orange (11 squares), amber (12 squares), khaki (19 squares), blue (20 squares), teal (24 squares), slate (28 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the slate region is the roomiest area on the grid with 28 squares.
Every color takes one queen and one only, whatever its size — which is why the 27 square difference between indigo and slate is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
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 khaki the most diffuse at 39% of its 7×7 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 12–12). It never leaves row 1, so its queen is locked to that row.
Sea green occupies 3 squares within rows 10–12, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Rose: 3 squares across rows 11–12, columns 11–12.
Red covers 4 squares, spanning 2 rows and 3 columns (rows 5–6, columns 2–4).
Mint occupies 9 squares within rows 3–6, columns 3–6.
Pale yellow: 10 squares across rows 2–7, columns 1–2.
Orange covers 11 squares, spanning 4 rows and 5 columns (rows 7–10, columns 1–5).
Amber occupies 12 squares within rows 1–3, columns 1–6.
Khaki: 19 squares across rows 5–11, columns 4–10.
Blue covers 20 squares, spanning 5 rows and 7 columns (rows 1–5, columns 6–12).
Teal occupies 24 squares within rows 9–12, columns 2–11.
Slate: 28 squares across rows 3–10, columns 7–12.
Where to start on this board
Indigo is the natural opening on #262: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 12–12. 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. Red 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 #262
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 8 and 12 are the narrowest, touched by only 3 colors. Columns 1, 7, 8, 9, 10, 11 and 12 are the tightest columns, with 4 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 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
The middle of #262 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.
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.
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 #262 is rated hard
Ratings track how generous the opening is. On #262 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 27-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
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.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #262 is row 1.
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 #262?
Puzzle #262 is rated hard. 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 #262 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #262, 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 #262?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.