12x12 Queens Puzzle #51
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 #51
What follows is a complete solve of puzzle #51, 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 pairing argument on columns.
Techniques this board needs: last square in a region, region shadow, pairing argument on columns, region confined to one line, pairing argument on rows, last square in a row.
Show the full walkthrough (18 steps) — contains the solution
Step 1. Only row 11, column 12 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 12, the rest of the teal region and every square adjacent to it — diagonals included.
Step 2. Mint is down to 3 candidates in rows 3 and 4, columns 7 and 8, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about mint itself.
Step 3. The pale yellow region has 3 squares left, spread across rows 7 and 8, columns 1 and 2. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.
Step 4. Here is the pairing argument. The slate and pale yellow regions have no open squares outside columns 1 and 2 — 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: 8 squares come off in one stroke.
Step 5. Only row 12, column 3 is still open in rose, and since each colour must take a queen, that settles it. That clears row 12, column 3, the rest of the rose region and every square adjacent to it — diagonals included.
The board after step 5 — 2 of 12 queens placed. Step 6. The amber region is down to one open square, at row 2, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 4, the remaining amber squares, and the ring of neighbours around row 2, column 4.
Step 7. 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: 4 squares in all.
Step 8. Blue is down to 3 candidates in rows 3 and 4, columns 6, 9, and 10, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about blue itself.
Step 9. The mint region has 2 squares left, spread across rows 3 and 4, columns 7 and 8. 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.
The board after step 9 — 3 of 12 queens placed. Step 10. Here is the pairing argument. The blue and mint regions have no open squares outside rows 3 and 4 — 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: 7 squares come off in one stroke.
Step 11. Red is down to 5 candidates in rows 5 and 6, columns 9–11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 12. Here is the pairing argument. The slate and red regions have no open squares outside rows 5 and 6 — 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: 7 squares come off in one stroke.
Step 13. Only row 8, column 5 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 5, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 14. The pale yellow region is down to one open square, at row 7, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 2, the remaining pale yellow squares, and the ring of neighbours around row 7, column 2.
The board after step 14 — 5 of 12 queens placed. Step 15. Only row 5, column 1 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 1, in slate and in the ring around that square is now dead.
Step 16. Wherever orange 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 17. Indigo, mint, and orange are now boxed into columns 6–8 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 1 square at once.
Step 18. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: row 4 is down to one square, putting a queen on row 4, column 7; indigo has one square left, so its queen takes row 1, column 6; orange has one square left, so its queen takes row 9, column 8; 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 9; red has one square left, so its queen takes row 6, column 11. 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 11.
The board after step 18 — 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 #51
12x12 puzzle #51 is rated hard. 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, #51 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 #51 are: teal (1 square), mint (3 squares), pale yellow (3 squares), indigo (9 squares), amber (9 squares), rose (9 squares), slate (10 squares), blue (15 squares), red (18 squares), orange (20 squares), khaki (23 squares), sea green (24 squares). An even split would hand every color 12 squares, so the teal region is running 11 squares under that average while the sea green region is the roomiest area on the grid with 24 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Teal has only 1 place to put its queen; sea green has 24. The spread between them on this board is 23 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 teal is the most solid, filling 100% of its 1×1 box, and khaki the most diffuse at 37% of its 9×7 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #51 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.
Teal: 1 square across rows 11–11, columns 12–12. It never leaves row 11, so its queen is locked to that row.
Mint covers 3 squares, spanning 2 rows and 2 columns (rows 3–4, columns 7–8).
Pale yellow occupies 3 squares within rows 7–8, columns 1–2.
Indigo: 9 squares across rows 1–5, columns 4–7.
Amber covers 9 squares, spanning 5 rows and 4 columns (rows 1–5, columns 1–4).
Rose occupies 9 squares within rows 9–12, columns 1–3.
Slate: 10 squares across rows 2–7, columns 1–2.
Blue covers 15 squares, spanning 4 rows and 7 columns (rows 1–4, columns 6–12).
Red occupies 18 squares within rows 2–9, columns 8–12.
Orange: 20 squares across rows 7–12, columns 5–10.
Khaki covers 23 squares, spanning 9 rows and 7 columns (rows 4–12, columns 2–8).
Sea green occupies 24 squares within rows 6–12, columns 7–12.
Where to start on this board
Begin where the choices are fewest. On this board that means teal, which owns just 1 square inside a 1×1 window running from row 11 to row 11 and column 12 to column 12. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because teal never leaves row 11, its queen must be in that row — and that instantly settles row 11 for everyone else. Sweep along it and X out every square that is not teal. 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. Mint and pale yellow 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 #51
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 3 colors. Columns 3, 4, 5 and 11 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 4, meeting 7 colors, is the opposite case — the busiest line on #51 and the last one you should expect to crack.
Working through the middle game
The middle of #51 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 #51 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 #51 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 23. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a hard board from the rest.
If #51 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 #51 the place to look for one is around row 1.
Here is a shape check that gives nothing away: when #51 is solved, 4 queens sit on the outer edge of the grid and 8 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 #51?
Puzzle #51 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #51 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #51, 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 #51?
Start with the teal region — the most constrained color on this board, with 1 square spanning rows 11–11 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.