12x12 Queens Puzzle #241

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 18-step reasoning is written out below.

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Step-by-step solution for puzzle #241

What follows is a complete solve of puzzle #241, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 12 queen placements and 6 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 rows.

Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, last square in a row, line confined to one region, last square in a column.

Show the full walkthrough (10 steps) — contains the solution

  1. Step 1. Only row 3, column 8 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 8, the rest of the blue region and every square adjacent to it — diagonals included.

  2. Step 2. 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. Cross out the whole of row 7 and column 11, the remaining pale yellow squares, and the ring of neighbours around row 7, column 11.

  3. 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: 9 squares in all.

    Puzzle #241 after step 3: 2 of 12 queens placed
    The board after step 3 — 2 of 12 queens placed.

  4. Step 4. Wherever orange 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.

  5. Step 5. Amber, mint, slate, and red are now boxed into rows 2, 4, 5, and 6 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 8 squares at once.

  6. Step 6. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: khaki has one square left, so its queen takes row 8, column 5; orange has one square left, so its queen takes row 9, column 1; amber has one square left, so its queen takes row 2, column 4; row 6 is down to one square, putting a queen on row 6, column 7; indigo has one square left, so its queen takes row 1, column 6; row 5 is down to one square, putting a queen on row 5, column 12. 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 12.

  7. Step 7. The teal region has 3 squares left, spread across rows 11 and 12, columns 2 and 3. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.

    Puzzle #241 after step 7: 8 of 12 queens placed
    The board after step 7 — 8 of 12 queens placed.

  8. Step 8. Sea green cannot leave row 10 — every one of its open squares (2 open squares) sits there. That row belongs to sea green, so every square on it owned by another colour is dead: 1 square in all.

  9. Step 9. Every open square left in column 10 (2 open squares) belongs to the rose region. Since that column must contain a queen, rose is the colour that supplies it — so rose squares anywhere else on the board are out, 2 squares in all.

  10. Step 10. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: column 9 is down to one square, putting a queen on row 10, column 9; rose has one square left, so its queen takes row 12, column 10; 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. 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 3.

    Puzzle #241 after step 10: 12 of 12 queens placed
    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 #241

Puzzle #241 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 #241, 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 #241 are: blue (1 square), pale yellow (1 square), orange (3 squares), indigo (8 squares), red (8 squares), mint (11 squares), rose (13 squares), amber (15 squares), khaki (16 squares), teal (18 squares), sea green (24 squares), slate (26 squares). An even split would hand every color 12 squares, so the blue region is running 11 squares under that average while the slate region is the roomiest area on the grid with 26 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 slate with 26. A 25-square gap like this one rewards players who start where the choices are fewest.

Square count is only half the story — how a color is arranged counts too. Blue is the tidiest shape on #241, packing 100% of its 1×1 bounding box, whereas sea green straggles across its 7×9 box at just 38%. 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.

Blue covers 1 square, spanning 1 row and 1 column (rows 3–3, columns 8–8). It never leaves row 3, so its queen is locked to that row.

Pale yellow occupies 1 square within rows 7–7, columns 11–11. It never leaves row 7, so its queen is locked to that row.

Orange: 3 squares across rows 8–9, columns 1–2.

Indigo covers 8 squares, spanning 4 rows and 3 columns (rows 1–4, columns 6–8).

Red occupies 8 squares within rows 4–6, columns 6–10.

Mint: 11 squares across rows 2–6, columns 1–5.

Rose covers 13 squares, spanning 4 rows and 5 columns (rows 9–12, columns 8–12).

Amber occupies 15 squares within rows 1–4, columns 1–6.

Khaki: 16 squares across rows 5–8, columns 1–8.

Teal covers 18 squares, spanning 5 rows and 6 columns (rows 8–12, columns 1–6).

Sea green occupies 24 squares within rows 6–12, columns 3–11.

Slate: 26 squares across rows 1–8, columns 9–12.

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 8 and 8. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #241.

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. Pale yellow and orange 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 #241

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 10, 11 and 12 are the narrowest, touched by only 3 colors. Column 12 is the tightest column, with 2 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 8, meeting 5 colors, is the opposite case — the busiest line on #241 and the last one you should expect to crack.

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.

The whole middle game reduces to two sweeps. Sweep the twelve colors looking for one with a single square left; sweep the twelve rows and twelve columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 12x12 solve from a stalled one.

The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.

Why puzzle #241 is rated hard

Ratings track how generous the opening is. On #241 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 25-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 #241 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 #241, start hunting for it near column 12.

Here is a shape check that gives nothing away: when #241 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

  1. The board is an N×N grid divided into N colored regions.
  2. Place exactly one queen (♛) in each row, each column, and each colored region.
  3. No two queens may touch each other — not horizontally, vertically, or diagonally.
  4. 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.
  5. 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 #241?

Puzzle #241 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 #241 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #241, 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 #241?

Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.