12x12 Queens Puzzle #292
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 #292
What follows is a complete solve of puzzle #292, 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, line confined to one region, pairing argument on rows, pairing argument on columns, proof by contradiction, last square in a row.
Show the full walkthrough (25 steps) — contains the solution
Step 1. Only row 6, column 10 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 10, the rest of the red region and every square adjacent to it — diagonals included.
Step 2. Indigo is down to 5 candidates in rows 1–3, columns 9, 11, and 12, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. The amber region has 3 squares left, spread across rows 1 and 2, columns 8 and 9. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 4. All of the mint 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: 10 squares come off.
Step 5. Only row 1, column 11 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 11, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 6. The amber region is down to one open square, at row 2, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 8, the remaining amber squares, and the ring of neighbours around row 2, column 8.
The board after step 6 — 3 of 12 queens placed. Step 7. Only row 8, column 9 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 9, in khaki and in the ring around that square is now dead.
Step 8. Sea green is down to 3 candidates in rows 10 and 11, columns 1 and 2, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about sea green itself.
Step 9. Teal cannot leave column 7 — every one of its open squares (2 open squares) sits there. That column belongs to teal, so every square in it owned by another colour is dead: 6 squares in all.
Step 10. Wherever slate 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 11. Row 12 has only rose squares left (6 open squares). The row needs a queen and only rose can provide one, which pins that colour to this row and kills its squares elsewhere: 8 squares.
Step 12. Here is the pairing argument. The blue, mint, and slate regions have no open squares outside rows 3–5 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 6 squares come off in one stroke.
Step 13. Pale yellow cannot leave row 7 — every one of its open squares (3 open squares) sits there. That row belongs to pale yellow, so every square on it owned by another colour is dead: 3 squares in all.
The board after step 13 — 4 of 12 queens placed. Step 14. Here is the pairing argument. The blue, pale yellow, orange, and sea green regions have no open squares outside columns 1–4 — 4 colours confined to 4 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 4 squares come off in one stroke.
Step 15. Test row 3, column 4 for blue: place a queen there, run the forced consequences through, and column 2 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 3. Follow the forced moves that would create and column 2 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 17. Test row 4, column 4 for blue: place a queen there, run the forced consequences through, and row 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 18. Row 4 has a single open square left, at column 12 in the mint region. Every row needs a queen, so this one is forced. Row 4 and column 12 are now spoken for, along with the squares touching that queen.
Step 19. Test row 5, column 3 for blue: place a queen there, run the forced consequences through, and column 2 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 19 — 5 of 12 queens placed. Step 20. Now a short proof by contradiction. Suppose the blue queen went to row 5, column 4. 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. Only row 3, column 3 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 3, the rest of the blue region and every square adjacent to it — diagonals included.
Step 22. 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 23. Only row 12, column 5 is still open in rose, and since each colour must take a queen, that settles it. Everything else in row 12, in column 5, in rose and in the ring around that square is now dead.
Step 24. Now a short proof by contradiction. Suppose the pale yellow queen went to row 7, column 4. Follow the forced moves that would create and row 10 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 25. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: pale yellow has one square left, so its queen takes row 7, column 1; sea green has one square left, so its queen takes row 10, column 2; orange has one square left, so its queen takes row 9, column 4; teal has one square left, so its queen takes row 11, column 7. 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 11, column 7.
The board after step 25 — 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 #292
This is puzzle number 292 in the 12x12 collection, rated medium. The grid holds 144 squares split between twelve colored regions, and a finished board carries twelve queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
As with every puzzle in this collection, #292 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 #292 are: red (1 square), amber (3 squares), mint (3 squares), sea green (3 squares), indigo (7 squares), orange (11 squares), khaki (13 squares), pale yellow (14 squares), rose (18 squares), blue (19 squares), teal (22 squares), slate (30 squares). An even split would hand every color 12 squares, so the red region is running 11 squares under that average while the slate region is the roomiest area on the grid with 30 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Red has only 1 place to put its queen; slate has 30. The spread between them on this board is 29 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Shape matters as much as size. The mint region is the most compact color here, filling 100% of the 3×1 box that encloses it, while pale yellow is the most scattered at 50% of its 7×4 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
Below is every color on #292 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.
Red covers 1 square, spanning 1 row and 1 column (rows 6–6, columns 10–10). It never leaves row 6, so its queen is locked to that row.
Amber occupies 3 squares within rows 1–2, columns 8–9.
Mint: 3 squares across rows 4–6, columns 12–12. It sits wholly inside column 12, so its queen must be in that column.
Sea green covers 3 squares, spanning 2 rows and 2 columns (rows 10–11, columns 1–2).
Indigo occupies 7 squares within rows 1–3, columns 9–12.
Orange: 11 squares across rows 7–10, columns 2–6.
Khaki covers 13 squares, spanning 5 rows and 5 columns (rows 6–10, columns 8–12).
Pale yellow occupies 14 squares within rows 3–9, columns 1–4.
Rose: 18 squares across rows 9–12, columns 1–7.
Blue covers 19 squares, spanning 5 rows and 7 columns (rows 1–5, columns 1–7).
Teal occupies 22 squares within rows 8–12, columns 6–12.
Slate: 30 squares across rows 1–7, columns 4–11.
Where to start on this board
The most constrained color is where a solve should begin, and on #292 that is red with 1 square. It reaches from row 6 to row 6 and from column 10 to column 10 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
Because red never leaves row 6, its queen must be in that row — and that instantly settles row 6 for everyone else. Sweep along it and X out every square that is not red. 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. Amber and sea green 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 #292
A quick way to find the next move is to count how many different colors touch each line. On this board, row 12 is the narrowest, touched by only 2 colors. Columns 1, 3, 5, 7, 8, 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 2 colors, so one of that handful must take the row and the rest are squeezed into the other 11 rows. Row 10 sits at the far end with 5 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.
Two checks, run alternately, will carry you through the 144 squares of this grid. First: has any of the twelve regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 12x12 boards almost always run one of those checks and forget the other.
Resist the plausible-looking queen. On #292 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 #292 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers three single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 29. 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.
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 #292 the place to look for one is around row 12.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 8 queens in the middle, and nothing in the corners. 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 #292?
Puzzle #292 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #292 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #292, 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 #292?
Start with the red region — the most constrained color on this board, with 1 square spanning rows 6–6 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.