12x12 Queens Puzzle #276
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 24-step reasoning is written out below.
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Step-by-step solution for puzzle #276
What follows is a complete solve of puzzle #276, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 12 queen placements and 12 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, region shadow, pairing argument on rows, proof by contradiction, line confined to one region, last square in a column.
Show the full walkthrough (18 steps) — contains the solution
Step 1. Only row 12, column 1 is still open in rose, and since each colour must take a queen, that settles it. That clears row 12, column 1, the rest of the rose region and every square adjacent to it — diagonals included.
Step 2. The teal region is down to one open square, at row 11, column 11. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 11, the remaining teal squares, and the ring of neighbours around row 11, column 11.
Step 3. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 10 squares in all.
Step 4. All of the sea green region's remaining squares (2 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 7 squares come off.
Step 5. Blue is down to 3 candidates in rows 2 and 3, columns 3 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
The board after step 5 — 2 of 12 queens placed. Step 6. Here is the pairing argument. The amber, blue, and mint regions have no open squares outside rows 2–4 — 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: 11 squares come off in one stroke.
Step 7. Slate cannot leave column 3 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 9 squares in all.
Step 8. The blue region is down to one open square, at row 3, column 4. Every colour holds exactly one queen, so that square is forced. Row 3 and column 4 are now spoken for, along with the squares touching that queen.
Step 9. Only row 1, column 5 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 5, the rest of the indigo region and every square adjacent to it — diagonals included.
The board after step 9 — 4 of 12 queens placed. Step 10. The sea green region is down to one open square, at row 10, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 2, the remaining sea green squares, and the ring of neighbours around row 10, column 2.
Step 11. Mint cannot leave row 4 — every one of its open squares (2 open squares) sits there. That row belongs to mint, so every square on it owned by another colour is dead: 3 squares in all.
Step 12. All of the red region's remaining squares (2 open squares) lie in column 6. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
Step 13. Wherever orange 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 14. Here is the pairing argument. The slate and red regions have no open squares outside rows 5 and 6 — 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: 7 squares come off in one stroke.
The board after step 14 — 5 of 12 queens placed. Step 15. Test row 2, column 9 for amber: 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 16. Every open square left in column 9 (2 open squares) belongs to the orange region. Since that column must contain a queen, orange is the colour that supplies it — so orange squares anywhere else on the board are out, 1 square in all.
Step 17. Orange cannot leave column 9 — 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: 3 squares in all.
Step 18. 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: column 10 is down to one square, putting a queen on row 7, column 10; orange has one square left, so its queen takes row 9, column 9; khaki has one square left, so its queen takes row 8, column 12; amber has one square left, so its queen takes row 2, column 8; mint has one square left, so its queen takes row 4, column 7; red has one square left, so its queen takes row 6, column 6; slate has one square left, so its queen takes row 5, column 3. 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 5, column 3.
The board after step 18 — 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 #276
This is puzzle number 276 in the 12x12 collection, rated hard. 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.
There is exactly one way to finish #276, 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 #276 are: rose (1 square), indigo (2 squares), mint (4 squares), teal (4 squares), slate (9 squares), khaki (12 squares), blue (13 squares), sea green (13 squares), amber (14 squares), red (15 squares), pale yellow (21 squares), orange (36 squares). An even split would hand every color 12 squares, so the rose region is running 11 squares under that average while the orange region is the roomiest area on the grid with 36 squares.
Every color takes one queen and one only, whatever its size — which is why the 35 square difference between rose and orange is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #276, packing 100% of its 1×2 bounding box, whereas red straggles across its 7×5 box at just 43%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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 hard board.
Rose: 1 square across rows 12–12, columns 1–1. It never leaves row 12, so its queen is locked to that row.
Indigo covers 2 squares, spanning 1 row and 2 columns (rows 1–1, columns 4–5). It never leaves row 1, so its queen is locked to that row.
Mint occupies 4 squares within rows 3–4, columns 7–8.
Teal: 4 squares across rows 11–12, columns 10–12.
Slate covers 9 squares, spanning 4 rows and 5 columns (rows 3–6, columns 1–5).
Khaki occupies 12 squares within rows 5–11, columns 10–12.
Blue: 13 squares across rows 1–4, columns 1–5.
Sea green covers 13 squares, spanning 7 rows and 4 columns (rows 6–12, columns 1–4).
Amber occupies 14 squares within rows 1–4, columns 7–12.
Red: 15 squares across rows 1–7, columns 4–8.
Pale yellow covers 21 squares, spanning 8 rows and 6 columns (rows 2–9, columns 7–12).
Orange occupies 36 squares within rows 7–12, columns 2–10.
Where to start on this board
The most constrained color is where a solve should begin, and on #276 that is rose with 1 square. It reaches from row 12 to row 12 and from column 1 to column 1 — 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 rose never leaves row 12, its queen must be in that row — and that instantly settles row 12 for everyone else. Sweep along it and X out every square that is not rose. 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, mint and teal 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 #276
A quick way to find the next move is to count how many different colors touch each line. On this board, row 10 is the narrowest, touched by only 3 colors. Column 6 is the tightest column, with 2 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 4 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
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.
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.
Resist the plausible-looking queen. On #276 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 #276 is rated hard
The hard label comes out of four measurable things about #276: three of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 35. Boards that score tight on those measures give up their first queen fast.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
Stuck on #276 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.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #276 is column 6.
As a sanity check rather than a spoiler: the finished solution to #276 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 #276?
Puzzle #276 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 #276 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #276, 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 #276?
Start with the rose region — the most constrained color on this board, with 1 square spanning rows 12–12 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.