12x12 Queens Puzzle #243
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 30-step reasoning is written out below.
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Step-by-step solution for puzzle #243
What follows is a complete solve of puzzle #243, worked the way a person works it rather than the way a computer does. It runs to 30 deductions — 12 queen placements and 18 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 columns, proof by contradiction, pairing argument on rows.
Show the full walkthrough (23 steps) — contains the solution
Step 1. Only row 7, column 5 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 5, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 2. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 11 squares come off.
Step 3. The khaki region has 2 squares left, spread across rows 6 and 8, columns 1 and 3. 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. All of the rose region's remaining squares (3 open squares) lie in row 12. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 9 squares come off.
Step 5. Teal is down to 6 candidates in rows 8–11, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about teal itself.
Step 6. Here is the pairing argument. The mint, khaki, and teal regions have no open squares outside columns 1–3 — 3 colours confined to 3 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.
The board after step 6 — 1 of 12 queens placed. Step 7. Amber cannot leave column 4 — every one of its open squares (4 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 3 squares in all.
Step 8. Sea green is down to 3 candidates in rows 10 and 11, columns 6 and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about sea green itself.
Step 9. Indigo and sea green are now boxed into columns 6 and 7 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 11 squares at once.
Step 10. Wherever red 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 11. Rose cannot leave row 12 — every one of its open squares (2 open squares) sits there. That row belongs to rose, so every square on it owned by another colour is dead: 1 square in all.
Step 12. Here is the pairing argument. The red and rose 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: 11 squares come off in one stroke.
The board after step 12 — 1 of 12 queens placed. Step 13. Wherever blue ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 14. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 6. Follow the forced moves that would create and the blue region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 15. Only row 1, column 7 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 7, in indigo and in the ring around that square is now dead.
Step 16. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 4. Follow the forced moves that would create and the slate region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 17. Test row 4, column 4 for amber: place a queen there, run the forced consequences through, and the mint region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 17 — 2 of 12 queens placed. Step 18. Now a short proof by contradiction. Suppose the amber queen went to row 5, column 4. Follow the forced moves that would create and column 3 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 19. Only row 2, column 4 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 4, in amber and in the ring around that square is now dead.
Step 20. The blue region is down to one open square, at row 3, column 10. Every colour holds exactly one queen, so that square is forced. Row 3 and column 10 are now spoken for, along with the squares touching that queen.
Step 21. Slate cannot leave column 12 — 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: 6 squares in all.
Step 22. Here is the pairing argument. The mint, slate, and red regions have no open squares outside rows 4–6 — 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: 2 squares come off in one stroke.
Step 23. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: khaki has one square left, so its queen takes row 8, column 1; teal has one square left, so its queen takes row 11, column 2; mint has one square left, so its queen takes row 4, column 3; slate has one square left, so its queen takes row 5, column 12; red has one square left, so its queen takes row 6, column 8; sea green has one square left, so its queen takes row 10, column 6; orange has one square left, so its queen takes row 9, column 11; rose has one square left, so its queen takes row 12, 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 12, column 9.
The board after step 23 — 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 #243
12x12 puzzle #243 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 #243, 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 #243 are: pale yellow (1 square), indigo (3 squares), rose (3 squares), mint (9 squares), slate (10 squares), khaki (10 squares), teal (10 squares), blue (15 squares), amber (16 squares), sea green (19 squares), red (20 squares), orange (28 squares). An even split would hand every color 12 squares, so the pale yellow region is running 11 squares under that average while the orange region is the roomiest area on the grid with 28 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Pale yellow has only 1 place to put its queen; orange has 28. The spread between them on this board is 27 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×3 box, and khaki the most diffuse at 40% of its 5×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
Pale yellow: 1 square across rows 7–7, columns 5–5. It never leaves row 7, so its queen is locked to that row.
Indigo covers 3 squares, spanning 1 row and 3 columns (rows 1–1, columns 5–7). It never leaves row 1, so its queen is locked to that row.
Rose occupies 3 squares within rows 12–12, columns 7–9. It never leaves row 12, so its queen is locked to that row.
Mint: 9 squares across rows 3–6, columns 1–3.
Slate covers 10 squares, spanning 5 rows and 4 columns (rows 3–7, columns 9–12).
Khaki occupies 10 squares within rows 4–8, columns 1–5.
Teal: 10 squares across rows 8–12, columns 1–3.
Blue covers 15 squares, spanning 4 rows and 5 columns (rows 1–4, columns 8–12).
Amber occupies 16 squares within rows 1–5, columns 1–6.
Sea green: 19 squares across rows 8–12, columns 2–8.
Red covers 20 squares, spanning 8 rows and 4 columns (rows 2–9, columns 6–9).
Orange occupies 28 squares within rows 5–12, columns 7–12.
Where to start on this board
Begin where the choices are fewest. On this board that means pale yellow, which owns just 1 square inside a 1×1 window running from row 7 to row 7 and column 5 to column 5. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because pale yellow never leaves row 7, its queen must be in that row — and that instantly settles row 7 for everyone else. Sweep along it and X out every square that is not pale yellow. 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 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 #243
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 10 and 11 are the narrowest, touched by only 3 colors. Columns 4, 10, 11 and 12 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 5, meeting 6 colors, is the opposite case — the busiest line on #243 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.
Resist the plausible-looking queen. On #243 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 #243 is rated hard
Ratings track how generous the opening is. On #243 you get four regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 27-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a hard board from the rest.
Stuck on #243 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 #243 the place to look for one is around row 1.
Here is a shape check that gives nothing away: when #243 is solved, 4 queens sit on the outer edge of the grid and 8 sit inside it, and no queen lands 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 #243?
Puzzle #243 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #243 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #243, 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 #243?
Start with the pale yellow region — the most constrained color on this board, with 1 square spanning rows 7–7 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.