12x12 Queens Puzzle #275
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 20-step reasoning is written out below.
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Step-by-step solution for puzzle #275
What follows is a complete solve of puzzle #275, worked the way a person works it rather than the way a computer does. It runs to 20 deductions — 12 queen placements and 8 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 confined to one line, line confined to one region, pairing argument on columns, region shadow, last square in a row.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Only row 10, column 2 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 2, the rest of the sea green region and every square adjacent to it — diagonals included.
Step 2. The rose region is down to one open square, at row 12, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 12 and column 1, the remaining rose squares, and the ring of neighbours around row 12, column 1.
Step 3. Mint cannot leave column 12 — every one of its open squares (3 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 8 squares in all.
Step 4. 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: 11 squares come off.
Step 5. Row 11 has only teal squares left (8 open squares). The row needs a queen and only teal can provide one, which pins that colour to this row and kills its squares elsewhere: 6 squares.
The board after step 5 — 2 of 12 queens placed. Step 6. Here is the pairing argument. The amber and pale yellow 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: 8 squares come off in one stroke.
Step 7. Red cannot leave column 9 — every one of its open squares (2 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 8 squares in all.
Step 8. The pale yellow region is down to one open square, at row 7, column 11. Every colour holds exactly one queen, so that square is forced. Row 7 and column 11 are now spoken for, along with the squares touching that queen.
Step 9. Orange cannot leave column 8 — every one of its open squares (4 open squares) sits there. That column belongs to orange, so every square in it owned by another colour is dead: 5 squares in all.
The board after step 9 — 3 of 12 queens placed. Step 10. The indigo region is down to one open square, at row 1, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 7, the remaining indigo squares, and the ring of neighbours around row 1, column 7.
Step 11. Only row 2, column 10 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 10, in amber and in the ring around that square is now dead.
Step 12. The slate region has 4 squares left, spread across rows 5 and 6, columns 4–6. 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. Column 3 has only khaki squares left (4 open squares). The column needs a queen and only khaki can provide one, which pins that colour to this column and kills its squares elsewhere: 4 squares.
Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: row 9 is down to one square, putting a queen on row 9, column 8; red has one square left, so its queen takes row 6, column 9; slate has one square left, so its queen takes row 5, column 4; mint has one square left, so its queen takes row 4, column 12; blue has one square left, so its queen takes row 3, column 6; khaki has one square left, so its queen takes row 8, column 3; teal has one square left, so its queen takes row 11, column 5. 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 5.
The board after step 14 — 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 #275
12x12 puzzle #275 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.
There is exactly one way to finish #275, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these twelve regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #275 are: sea green (1 square), mint (3 squares), rose (6 squares), orange (7 squares), amber (8 squares), slate (8 squares), indigo (9 squares), pale yellow (10 squares), red (14 squares), blue (23 squares), khaki (24 squares), teal (31 squares). An even split would hand every color 12 squares, so the sea green region is running 11 squares under that average while the teal region is the roomiest area on the grid with 31 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Sea green has only 1 place to put its queen; teal has 31. The spread between them on this board is 30 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 teal is the most scattered at 52% of its 6×10 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 #275 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 1 square within rows 10–10, columns 2–2. It never leaves row 10, so its queen is locked to that row.
Mint: 3 squares across rows 4–6, columns 12–12. It sits wholly inside column 12, so its queen must be in that column.
Rose covers 6 squares, spanning 3 rows and 3 columns (rows 10–12, columns 1–3).
Orange occupies 7 squares within rows 5–9, columns 8–9.
Amber: 8 squares across rows 1–3, columns 10–12.
Slate covers 8 squares, spanning 3 rows and 4 columns (rows 5–7, columns 4–7).
Indigo occupies 9 squares within rows 1–4, columns 7–9.
Pale yellow: 10 squares across rows 7–12, columns 10–12.
Red covers 14 squares, spanning 7 rows and 3 columns (rows 3–9, columns 9–11).
Blue occupies 23 squares within rows 1–5, columns 1–7.
Khaki: 24 squares across rows 3–10, columns 1–5.
Teal covers 31 squares, spanning 6 rows and 10 columns (rows 7–12, columns 2–11).
Where to start on this board
Begin where the choices are fewest. On this board that means sea green, which owns just 1 square inside a 1×1 window running from row 10 to row 10 and column 2 to column 2. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: orange spans only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #275
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 11 and 12 are the narrowest, touched by only 3 colors. Columns 1, 6, 8 and 12 are the tightest columns, with 3 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 11 rows. By contrast row 7 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
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.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #275 is rated hard
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 3 colors, and a region-size spread of 30. 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.
Stuck on #275 despite the hard rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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 #275 the place to look for one is around row 1.
As a sanity check rather than a spoiler: the finished solution to #275 puts 3 queens on the outer border and 9 queens in the interior, 1 of them in an actual corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 #275?
Puzzle #275 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 #275 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #275, 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 #275?
Start with the sea green region — the most constrained color on this board, with 1 square spanning rows 10–10 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.