12x12 Queens Puzzle #125
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 27-step reasoning is written out below.
Previous puzzle · All 12x12 puzzles · Next puzzle
Step-by-step solution for puzzle #125
What follows is a complete solve of puzzle #125, worked the way a person works it rather than the way a computer does. It runs to 27 deductions — 12 queen placements and 15 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: region shadow, proof by contradiction, last square in a region, region confined to one line, pairing argument on columns.
Show the full walkthrough (20 steps) — contains the solution
Step 1. Wherever indigo 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 2. Blue is down to 4 candidates in rows 1–3, columns 11 and 12, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 3. The orange region has 6 squares left, spread across rows 6–9, columns 3–5. 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 4. Wherever sea green ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 5. Test row 1, column 2 for indigo: 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 5 — 0 of 12 queens placed. Step 6. The indigo region has 3 squares left, spread across rows 1 and 2, columns 2–4. 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. Test row 1, column 3 for indigo: place a queen there, run the forced consequences through, and the pale yellow region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 8. Indigo is down to 2 candidates in rows 1 and 2, columns 2 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 9. Test row 2, column 2 for indigo: 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.
Step 10. The indigo region is down to one open square, at row 1, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 4, the remaining indigo squares, and the ring of neighbours around row 1, column 4.
The board after step 10 — 1 of 12 queens placed. Step 11. Blue cannot leave column 12 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 10 squares in all.
Step 12. The slate region has 3 squares left, spread across rows 3–5, columns 10 and 11. 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 13. Wherever orange ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 14. Here is the pairing argument. The mint, pale yellow, orange, and sea green regions have no open squares outside columns 1, 2, 3, and 5 — 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: 9 squares come off in one stroke.
Step 15. Only row 11, column 6 is still open in teal, and since each colour must take a queen, that settles it. Everything else in row 11, in column 6, in teal and in the ring around that square is now dead.
The board after step 15 — 2 of 12 queens placed. Step 16. The sea green region is down to one open square, at row 10, column 1. Every colour holds exactly one queen, so that square is forced. Row 10 and column 1 are now spoken for, along with the squares touching that queen.
Step 17. Amber cannot leave row 2 — every one of its open squares (3 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 2 squares in all.
Step 18. 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: blue has one square left, so its queen takes row 3, column 12; slate has one square left, so its queen takes row 5, column 11; mint has one square left, so its queen takes row 4, column 2; red has one square left, so its queen takes row 6, column 9. 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 9.
Step 19. All of the pale yellow region's remaining squares (2 open squares) lie in row 7. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 1 square come off.
Step 20. 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: khaki has one square left, so its queen takes row 8, column 8; orange has one square left, so its queen takes row 9, column 5; pale yellow has one square left, so its queen takes row 7, column 3; rose has one square left, so its queen takes row 12, column 10; amber has one square left, so its queen takes row 2, 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 2, column 7.
The board after step 20 — 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 #125
Puzzle #125 is one of the hard-rated boards in the 12x12 set. Twelve colors divide its 144 squares, and you win by placing twelve queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
Like every board on this site, #125 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 #125 are: sea green (3 squares), indigo (4 squares), blue (4 squares), orange (6 squares), slate (7 squares), mint (12 squares), khaki (14 squares), teal (16 squares), amber (17 squares), red (17 squares), pale yellow (17 squares), rose (27 squares). An even split would hand every color 12 squares, so the sea green region is running 9 squares under that average while the rose region is the roomiest area on the grid with 27 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Sea green is working with 3 squares while rose has 27 to spare, a gap of 24. Attack the cramped end and the roomy regions tend to resolve themselves.
Square count is only half the story — how a color is arranged counts too. Rose is the tidiest shape on #125, packing 77% of its 5×7 bounding box, whereas amber straggles across its 6×8 box at just 35%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Below is every color on #125 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.
Sea green occupies 3 squares within rows 10–11, columns 1–2.
Indigo: 4 squares across rows 1–2, columns 2–4.
Blue covers 4 squares, spanning 3 rows and 2 columns (rows 1–3, columns 11–12).
Orange occupies 6 squares within rows 6–9, columns 3–5.
Slate: 7 squares across rows 3–6, columns 10–12.
Mint covers 12 squares, spanning 5 rows and 5 columns (rows 1–5, columns 1–5).
Khaki occupies 14 squares within rows 6–10, columns 6–10.
Teal: 16 squares across rows 9–12, columns 1–6.
Amber covers 17 squares, spanning 6 rows and 8 columns (rows 1–6, columns 4–11).
Red occupies 17 squares within rows 2–7, columns 7–12.
Pale yellow: 17 squares across rows 4–9, columns 1–5.
Rose covers 27 squares, spanning 5 rows and 7 columns (rows 8–12, columns 6–12).
Where to start on this board
Open on the tightest color. Here that is sea green, confined to 3 squares between rows 10 and 11 and columns 1 and 2. Its queen has nowhere else to go, which means those 2 rows and 2 columns carry more constraint than anywhere else on #125.
Sea green does not collapse onto a single line, so it will not hand you a queen on move one. What it does give you is a short list of 3 candidates. Take each one in turn, imagine the queen there, cross out its row, its column and the eight squares around it, and ask whether any other region is left with nowhere to go. One or two candidates usually fail that test immediately.
Work outward to the other flat regions next. Indigo is no more than two rows tall, so its queen is 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 #125
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, 6, 7, 8, 9 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 6 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.
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.
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 #125 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers no single-line region at all, a 3-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 24. 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.
If #125 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.
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 row 12 on this board.
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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 12x12 Queens puzzle #125?
Puzzle #125 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 2 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 12x12 puzzle #125 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #125, 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 #125?
Start with the sea green region — the most constrained color on this board, with 3 squares spanning rows 10–11 and columns 1–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.