12x12 Queens Puzzle #272
Difficulty: easy. 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 26-step reasoning is written out below.
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Step-by-step solution for puzzle #272
What follows is a complete solve of puzzle #272, worked the way a person works it rather than the way a computer does. It runs to 26 deductions — 12 queen placements and 14 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, line confined to one region, last square in a row.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Only row 1, column 6 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 6, the rest of the indigo 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. The amber region has 4 squares left, spread across rows 2–4, columns 11 and 12. 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. Wherever red ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 4 — 2 of 12 queens placed. Step 5. Khaki is down to 4 candidates in rows 8–10, columns 2–4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 6. The sea green region has 2 squares left, spread across rows 10 and 11, columns 2 and 3. 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 7. Wherever khaki ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 8. Here is the pairing argument. The amber and slate regions have no open squares outside columns 11 and 12 — 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: 10 squares come off in one stroke.
Step 9. Orange cannot leave column 10 — every one of its open squares (2 open squares) sits there. That column belongs to orange, so every square in it owned by another colour is dead: 10 squares in all.
The board after step 9 — 2 of 12 queens placed. Step 10. Wherever mint ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. Khaki and orange are now boxed into rows 8 and 9 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 5 squares at once.
Step 12. Here is the pairing argument. The sea green and teal regions have no open squares outside rows 10 and 11 — 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: 2 squares come off in one stroke.
Step 13. Slate, red, and pale yellow are now boxed into rows 5–7 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 13 squares at once.
The board after step 13 — 2 of 12 queens placed. Step 14. All of the mint region's remaining squares (2 open squares) lie in row 4. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.
Step 15. Row 5 has only slate squares left (2 open squares). The row needs a queen and only slate can provide one, which pins that colour to this row and kills its squares elsewhere: 2 squares.
Step 16. Here is the pairing argument. The mint and pale yellow regions have no open squares outside columns 8 and 9 — 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: 6 squares come off in one stroke.
Step 17. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: row 10 is down to one square, putting a queen on row 10, column 2; khaki has one square left, so its queen takes row 8, column 4; 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 9; mint has one square left, so its queen takes row 4, column 8; blue has one square left, so its queen takes row 3, column 5; amber has one square left, so its queen takes row 2, column 12; slate has one square left, so its queen takes row 5, column 11; orange has one square left, so its queen takes row 9, column 10; 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 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 #272
This is puzzle number 272 in the 12x12 collection, rated easy. 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.
Puzzle #272 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 #272 are: indigo (1 square), rose (1 square), slate (5 squares), khaki (5 squares), mint (7 squares), amber (8 squares), red (8 squares), sea green (12 squares), orange (15 squares), teal (16 squares), pale yellow (17 squares), blue (49 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the blue region is the roomiest area on the grid with 49 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 1 square while blue has 49 to spare, a gap of 48. 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. Indigo is the tidiest shape on #272, packing 100% of its 1×1 bounding box, whereas pale yellow straggles across its 6×7 box at just 40%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Indigo occupies 1 square within rows 1–1, columns 6–6. It never leaves row 1, so its queen is locked to that row.
Rose: 1 square across rows 12–12, columns 1–1. It never leaves row 12, so its queen is locked to that row.
Slate covers 5 squares, spanning 3 rows and 2 columns (rows 4–6, columns 11–12).
Khaki occupies 5 squares within rows 8–10, columns 2–4.
Mint: 7 squares across rows 4–6, columns 7–10.
Amber covers 8 squares, spanning 4 rows and 4 columns (rows 1–4, columns 9–12).
Red occupies 8 squares within rows 6–8, columns 1–4.
Sea green: 12 squares across rows 9–12, columns 1–7.
Orange covers 15 squares, spanning 6 rows and 3 columns (rows 7–12, columns 10–12).
Teal occupies 16 squares within rows 9–12, columns 4–10.
Pale yellow: 17 squares across rows 5–10, columns 4–10.
Blue covers 49 squares, spanning 7 rows and 11 columns (rows 1–7, columns 1–11).
Where to start on this board
The most constrained color is where a solve should begin, and on #272 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 6 to column 6 — 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 indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Rose 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 #272
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2 and 3 are the narrowest, touched by only 2 colors. Column 12 is the tightest column, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 10, meeting 5 colors, is the opposite case — the busiest line on #272 and the last one you should expect to crack.
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.
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.
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 #272 is rated easy
Ratings track how generous the opening is. On #272 you get four regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 48-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.
Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.
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 row 2 on this board.
Here is a shape check that gives nothing away: when #272 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.
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Questions about this puzzle
How hard is 12x12 Queens puzzle #272?
Puzzle #272 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #272 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #272, 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 #272?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.