12x12 Queens Puzzle #251
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 25-step reasoning is written out below.
Previous puzzle · All 12x12 puzzles · Next puzzle
Step-by-step solution for puzzle #251
What follows is a complete solve of puzzle #251, worked the way a person works it rather than the way a computer does. It runs to 25 deductions — 12 queen placements and 13 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 columns, proof by contradiction, region shadow, last square in a row.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Only row 5, column 1 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 1, the rest of the slate region and every square adjacent to it — diagonals included.
Step 2. All of the indigo region's remaining squares (2 open squares) lie in column 12. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 11 squares come off.
Step 3. Red cannot leave column 3 — every one of its open squares (3 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 10 squares in all.
Step 4. All of the pale yellow region's remaining squares (2 open squares) lie in column 5. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 12 squares come off.
The board after step 4 — 1 of 12 queens placed. Step 5. Khaki cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 6 squares in all.
Step 6. Here is the pairing argument. The mint and rose regions have no open squares outside columns 4 and 6 — 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 7. Only row 2, column 7 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 7, in amber and in the ring around that square is now dead.
Step 8. The indigo region is down to one open square, at row 1, column 12. Every colour holds exactly one queen, so that square is forced. Row 1 and column 12 are now spoken for, along with the squares touching that queen.
Step 9. Test row 3, column 9 for blue: place a queen there, run the forced consequences through, and the teal region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 9 — 3 of 12 queens placed. Step 10. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 11. Follow the forced moves that would create and the sea green region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 11. Test row 4, column 8 for blue: 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 12. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 9. Follow the forced moves that would create and the teal region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 13. Test row 4, column 11 for blue: place a queen there, run the forced consequences through, and the sea green region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 13 — 3 of 12 queens placed. Step 14. Blue is down to 5 candidates in rows 3, 6, and 7, columns 8, 10, and 11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 15. Test row 6, column 8 for blue: place a queen there, run the forced consequences through, and the red region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 16. Here is the pairing argument. The blue and sea green regions have no open squares outside columns 10 and 11 — 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: 1 square come off in one stroke.
Step 17. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 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 4; rose has one square left, so its queen takes row 12, column 6; khaki has one square left, so its queen takes row 8, column 2; red has one square left, so its queen takes row 6, column 3; pale yellow has one square left, so its queen takes row 7, column 5; blue has one square left, so its queen takes row 3, column 10; orange has one square left, so its queen takes row 9, column 8; teal has one square left, so its queen takes row 11, column 9; sea green has one square left, so its queen takes row 10, 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 10, column 11.
The board after step 17 — 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 #251
12x12 puzzle #251 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 #251 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 #251 are: slate (1 square), indigo (2 squares), pale yellow (2 squares), red (8 squares), amber (9 squares), teal (10 squares), rose (12 squares), khaki (13 squares), sea green (16 squares), orange (19 squares), mint (24 squares), blue (28 squares). An even split would hand every color 12 squares, so the slate region is running 11 squares under that average while the blue region is the roomiest area on the grid with 28 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Slate is working with 1 square while blue has 28 to spare, a gap of 27. Attack the cramped end and the roomy regions tend to resolve themselves.
Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 2×1 box that encloses it, while red is the most scattered at 44% of its 6×3 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
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.
Slate occupies 1 square within rows 5–5, columns 1–1. It never leaves row 5, so its queen is locked to that row.
Indigo: 2 squares across rows 1–2, columns 12–12. It sits wholly inside column 12, so its queen must be in that column.
Pale yellow covers 2 squares, spanning 2 rows and 1 column (rows 7–8, columns 5–5). It sits wholly inside column 5, so its queen must be in that column.
Red occupies 8 squares within rows 3–8, columns 1–3.
Amber: 9 squares across rows 1–4, columns 4–7.
Teal covers 10 squares, spanning 4 rows and 4 columns (rows 9–12, columns 6–9).
Rose occupies 12 squares within rows 9–12, columns 2–6.
Khaki: 13 squares across rows 6–12, columns 1–3.
Sea green covers 16 squares, spanning 6 rows and 3 columns (rows 7–12, columns 10–12).
Orange occupies 19 squares within rows 4–10, columns 5–10.
Mint: 24 squares across rows 1–8, columns 1–6.
Blue covers 28 squares, spanning 7 rows and 6 columns (rows 1–7, columns 7–12).
Where to start on this board
Begin where the choices are fewest. On this board that means slate, which owns just 1 square inside a 1×1 window running from row 5 to row 5 and column 1 to column 1. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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 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 #251
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3, 11 and 12 are the narrowest, touched by only 4 colors. Column 11 is the tightest column, with 2 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 4 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 7, meeting 7 colors, is the opposite case — the busiest line on #251 and the last one you should expect to crack.
Working through the middle game
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
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.
Resist the plausible-looking queen. On #251 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 #251 is rated medium
The medium label comes out of four measurable things about #251: four of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 4 colors, and its largest and smallest regions differ by 27. Boards that score tight on those measures give up their first queen fast.
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 medium board from the rest.
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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around column 11 on this board.
Here is a shape check that gives nothing away: when #251 is solved, 3 queens sit on the outer edge of the grid and 9 sit inside it, with 1 queen 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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 12x12 Queens puzzle #251?
Puzzle #251 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 4 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #251 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #251, 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 #251?
Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.