12x12 Queens Puzzle #36

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 52-step reasoning is written out below.

Previous puzzle · All 12x12 puzzles · Next puzzle

Step-by-step solution for puzzle #36

What follows is a complete solve of puzzle #36, worked the way a person works it rather than the way a computer does. It runs to 52 deductions — 12 queen placements and 40 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, proof by contradiction, pairing argument on columns, last square in a row.

Show the full walkthrough (47 steps) — contains the solution

  1. Step 1. Only row 1, column 12 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 12, the rest of the indigo region and every square adjacent to it — diagonals included.

  2. Step 2. All of the amber region's remaining squares (2 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.

  3. Step 3. Blue cannot leave row 3 — every one of its open squares (3 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 9 squares in all.

  4. Step 4. All of the slate region's remaining squares (3 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 10 squares come off.

  5. Step 5. Orange is down to 6 candidates in rows 8 and 9, columns 5–9, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about orange itself.

  6. Step 6. The sea green region has 4 squares left, spread across rows 9 and 10, columns 9–11. 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.

  7. Step 7. Test row 4, column 5 for mint: place a queen there, run the forced consequences through, and the sea green region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.

  8. Step 8. Now a short proof by contradiction. Suppose the mint queen went to row 4, column 7. Follow the forced moves that would create and the sea green region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

  9. Step 9. Test row 4, column 8 for mint: place a queen there, run the forced consequences through, and the rose region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.

  10. Step 10. Now a short proof by contradiction. Suppose the mint queen went to row 4, column 9. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

  11. Step 11. Test row 4, column 10 for mint: place a queen there, run the forced consequences through, and the sea green region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.

  12. Step 12. The mint region has 3 squares left, spread across rows 4 and 6, columns 4 and 5. 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.

    Puzzle #36 after step 12: 1 of 12 queens placed
    The board after step 12 — 1 of 12 queens placed.

  13. Step 13. Slate cannot leave row 5 — every one of its open squares (2 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 1 square in all.

  14. Step 14. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 4. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

  15. Step 15. Blue cannot leave row 3 — every one of its open squares (2 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 1 square in all.

  16. Step 16. Wherever red 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.

  17. Step 17. Blue, red, and teal are now boxed into columns 1–3 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 7 squares at once.

  18. Step 18. Now a short proof by contradiction. Suppose the mint queen went to row 6, column 5. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

  19. Step 19. Mint cannot leave column 4 — every one of its open squares (2 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 5 squares in all.

  20. Step 20. Now a short proof by contradiction. Suppose the slate queen went to row 5, column 7. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

  21. Step 21. Only row 5, column 6 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 6, the rest of the slate region and every square adjacent to it — diagonals included.

  22. Step 22. Now a short proof by contradiction. Suppose the red queen went to row 4, column 1. Follow the forced moves that would create and column 5 runs out of open squares entirely — impossible. So that square is ruled out for good.

  23. Step 23. Test row 6, column 8 for pale yellow: place a queen there, run the forced consequences through, and the khaki region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.

  24. Step 24. Now a short proof by contradiction. Suppose the pale yellow queen went to row 6, column 9. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

    Puzzle #36 after step 24: 2 of 12 queens placed
    The board after step 24 — 2 of 12 queens placed.

  25. Step 25. Test row 6, column 10 for pale yellow: place a queen there, run the forced consequences through, and the khaki region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.

  26. Step 26. Now a short proof by contradiction. Suppose the pale yellow queen went to row 6, column 11. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

  27. Step 27. Test row 7, column 7 for pale yellow: place a queen there, run the forced consequences through, and the khaki region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.

  28. Step 28. Now a short proof by contradiction. Suppose the pale yellow queen went to row 7, column 8. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

  29. Step 29. Test row 7, column 9 for pale yellow: place a queen there, run the forced consequences through, and column 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.

  30. Step 30. Now a short proof by contradiction. Suppose the pale yellow queen went to row 7, column 10. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

  31. Step 31. Test row 7, column 11 for pale yellow: place a queen there, run the forced consequences through, and column 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.

  32. Step 32. Now a short proof by contradiction. Suppose the pale yellow queen went to row 9, column 5. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

  33. Step 33. Test row 10, column 5 for pale yellow: place a queen there, run the forced consequences through, and the sea green region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.

  34. Step 34. Now a short proof by contradiction. Suppose the pale yellow queen went to row 11, column 5. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

  35. Step 35. Test row 12, column 5 for pale yellow: place a queen there, run the forced consequences through, and the rose region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.

    Puzzle #36 after step 35: 2 of 12 queens placed
    The board after step 35 — 2 of 12 queens placed.

  36. Step 36. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 3. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

  37. Step 37. Only row 3, column 2 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 2, the rest of the blue region and every square adjacent to it — diagonals included.

  38. Step 38. All of the red region's remaining squares (2 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.

  39. Step 39. Test row 6, column 4 for mint: place a queen there, run the forced consequences through, and the rose region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.

  40. Step 40. The mint region is down to one open square, at row 4, column 4. Every colour holds exactly one queen, so that square is forced. Row 4 and column 4 are now spoken for, along with the squares touching that queen.

  41. Step 41. Look along row 6: one square survives, row 6, column 1, which belongs to red. A row cannot be left empty, so the queen goes there. Everything else in row 6, in column 1, in red and in the ring around that square is now dead.

  42. Step 42. Row 7 has a single open square left, at column 5 in the pale yellow region. Every row needs a queen, so this one is forced. Row 7 and column 5 are now spoken for, along with the squares touching that queen.

  43. Step 43. Khaki cannot leave row 8 — every one of its open squares (3 open squares) sits there. That row belongs to khaki, so every square on it owned by another colour is dead: 2 squares in all.

  44. Step 44. All of the orange region's remaining squares (2 open squares) lie in row 9. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 3 squares come off.

  45. Step 45. 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: 2 squares in all.

  46. Step 46. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 10. Follow the forced moves that would create and column 8 runs out of open squares entirely — impossible. So that square is ruled out for good.

  47. Step 47. 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: amber has one square left, so its queen takes row 2, column 7; orange has one square left, so its queen takes row 9, column 8; sea green has one square left, so its queen takes row 10, column 10; khaki has one square left, so its queen takes row 8, column 11; rose has one square left, so its queen takes row 12, column 9; teal has one square left, so its queen takes row 11, 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 11, column 3.

    Puzzle #36 after step 47: 12 of 12 queens placed
    The board after step 47 — 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 #36

This is puzzle number 36 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.

Puzzle #36 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 #36 are: indigo (1 square), slate (3 squares), sea green (5 squares), orange (6 squares), blue (8 squares), red (8 squares), teal (10 squares), amber (11 squares), khaki (14 squares), rose (14 squares), mint (24 squares), pale yellow (40 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the pale yellow region is the roomiest area on the grid with 40 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 1 place to put its queen; pale yellow has 40. The spread between them on this board is 39 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #36, packing 100% of its 1×1 bounding box, whereas pale yellow straggles across its 8×11 box at just 45%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Here is the full footprint of every color on puzzle #36, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Indigo: 1 square across rows 1–1, columns 12–12. It never leaves row 1, so its queen is locked to that row.

Slate covers 3 squares, spanning 1 row and 3 columns (rows 5–5, columns 5–7). It never leaves row 5, so its queen is locked to that row.

Sea green occupies 5 squares within rows 9–10, columns 9–12.

Orange: 6 squares across rows 8–9, columns 5–9.

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

Red occupies 8 squares within rows 3–7, columns 1–2.

Teal: 10 squares across rows 9–12, columns 1–3.

Amber covers 11 squares, spanning 2 rows and 9 columns (rows 1–2, columns 3–11).

Khaki occupies 14 squares within rows 2–8, columns 9–12.

Rose: 14 squares across rows 10–12, columns 7–12.

Mint covers 24 squares, spanning 5 rows and 8 columns (rows 2–6, columns 3–10).

Pale yellow occupies 40 squares within rows 5–12, columns 1–11.

Where to start on this board

The most constrained color is where a solve should begin, and on #36 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 12 to column 12 — 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. Slate, amber, orange and sea green 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 #36

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 4, 7, 8, 11 and 12 are the narrowest, touched by only 3 colors. Columns 1, 2, 4 and 12 are the tightest columns, with 4 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

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 #36 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 #36 is rated hard

The rating reflects how much the board concedes up front. Puzzle #36 has three regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 39 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

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.

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 #36 the place to look for one is around row 1.

As a sanity check rather than a spoiler: the finished solution to #36 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

  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.

Full rules · Strategy library · Puzzle statistics

Questions about this puzzle

How hard is 12x12 Queens puzzle #36?

Puzzle #36 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 #36 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #36, 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 #36?

Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.