12x12 Queens Puzzle #249
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 19-step reasoning is written out below.
Previous puzzle · All 12x12 puzzles · Next puzzle
Step-by-step solution for puzzle #249
What follows is a complete solve of puzzle #249, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 12 queen placements and 7 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, region shadow, pairing argument on rows, pairing argument on columns, last square in a column, last square in a row.
Show the full walkthrough (10 steps) — contains the solution
Step 1. 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 2. The amber region is down to one open square, at row 2, column 12. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 12, the remaining amber squares, and the ring of neighbours around row 2, column 12.
Step 3. Pale yellow cannot leave column 11 — every one of its open squares (3 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 7 squares in all.
The board after step 3 — 2 of 12 queens placed. Step 4. Wherever rose 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 5. Indigo, mint, and slate are now boxed into rows 1, 4, and 5 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 8 squares at once.
Step 6. All of the pale yellow region's remaining squares (2 open squares) lie in column 11. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
Step 7. Red and pale yellow are now boxed into rows 6 and 7 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 9 squares at once.
The board after step 7 — 2 of 12 queens placed. Step 8. Khaki is down to 5 candidates in rows 8–10, columns 4–6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 9. Slate, red, teal, and rose are now boxed into columns 1, 2, 4, and 5 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 8 squares at once.
Step 10. 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: khaki has one square left, so its queen takes row 8, column 6; indigo has one square left, so its queen takes row 1, column 7; column 5 is down to one square, putting a queen on row 6, column 5; pale yellow has one square left, so its queen takes row 7, column 11; row 5 is down to one square, putting a queen on row 5, column 2; column 4 is down to one square, putting a queen on row 11, column 4; rose has one square left, so its queen takes row 12, column 1; sea green has one square left, so its queen takes row 10, column 8; orange has one square left, so its queen takes row 9, column 10; mint has one square left, so its queen takes row 4, 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 4, column 9.
The board after step 10 — 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 #249
Puzzle #249 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.
There is exactly one way to finish #249, 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 #249 are: blue (1 square), amber (3 squares), rose (4 squares), pale yellow (7 squares), indigo (9 squares), khaki (10 squares), teal (12 squares), red (13 squares), sea green (15 squares), mint (18 squares), slate (19 squares), orange (33 squares). An even split would hand every color 12 squares, so the blue region is running 11 squares under that average while the orange region is the roomiest area on the grid with 33 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So blue, with 1 square, is under far more pressure than orange with 33. A 32-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Blue is the densest region on this grid at 100% of a 1×1 frame; khaki is the loosest, occupying only 50% of its 5×4 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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.
Blue: 1 square across rows 3–3, columns 3–3. It never leaves row 3, so its queen is locked to that row.
Amber covers 3 squares, spanning 2 rows and 2 columns (rows 2–3, columns 11–12).
Rose occupies 4 squares within rows 11–12, columns 1–3.
Pale yellow: 7 squares across rows 4–7, columns 11–12.
Indigo covers 9 squares, spanning 4 rows and 3 columns (rows 1–4, columns 5–7).
Khaki occupies 10 squares within rows 6–10, columns 4–7.
Teal: 12 squares across rows 8–11, columns 1–4.
Red covers 13 squares, spanning 4 rows and 6 columns (rows 4–7, columns 1–6).
Sea green occupies 15 squares within rows 9–12, columns 4–9.
Mint: 18 squares across rows 1–5, columns 7–12.
Slate covers 19 squares, spanning 5 rows and 5 columns (rows 1–5, columns 1–5).
Orange occupies 33 squares within rows 5–12, columns 7–12.
Where to start on this board
Open on the tightest color. Here that is blue, confined to 1 square between rows 3 and 3 and columns 3 and 3. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #249.
Because blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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 rose 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 #249
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 8 and 12 are the narrowest, touched by only 3 colors. Column 10 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 5 meets 5 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.
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.
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 #249 is rated hard
The hard label comes out of four measurable things about #249: two 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 32. Boards that score tight on those measures give up their first queen fast.
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 #249 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.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #249, start hunting for it near column 10.
As a sanity check rather than a spoiler: the finished solution to #249 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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 12x12 Queens puzzle #249?
Puzzle #249 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #249 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #249, 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 #249?
Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.