12x12 Queens Puzzle #180

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 25-step reasoning is written out below.

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Step-by-step solution for puzzle #180

What follows is a complete solve of puzzle #180, worked the way a person works it rather than the way a computer does. It runs to 25 deductions — 12 queen placements and 13 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.

Show the full walkthrough (18 steps) — contains the solution

  1. Step 1. Only row 5, column 7 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 7, the rest of the slate region and every square adjacent to it — diagonals included.

  2. Step 2. All of the mint region's remaining squares (2 open squares) lie in column 12. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 11 squares come off.

  3. Step 3. The amber region has 3 squares left, spread across rows 1 and 2, columns 10 and 11. 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.

  4. Step 4. 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.

  5. Step 5. Orange is down to 3 candidates in rows 8 and 9, columns 10 and 11, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about orange itself.

    Puzzle #180 after step 5: 1 of 12 queens placed
    The board after step 5 — 1 of 12 queens placed.

  6. Step 6. Here is the pairing argument. The indigo, amber, blue, and mint regions have no open squares outside rows 1–4 — 4 colours confined to 4 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 6 squares come off in one stroke.

  7. Step 7. Wherever red 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.

  8. Step 8. All of the khaki 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: 7 squares come off.

  9. Step 9. Red cannot leave row 6 — every one of its open squares (2 open squares) sits there. That row belongs to red, so every square on it owned by another colour is dead: 5 squares in all.

    Puzzle #180 after step 9: 1 of 12 queens placed
    The board after step 9 — 1 of 12 queens placed.

  10. Step 10. All of the pale yellow region's remaining squares (2 open squares) lie in row 7. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.

  11. Step 11. Only row 8, column 1 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 1, in khaki and in the ring around that square is now dead.

  12. Step 12. The orange region is down to one open square, at row 9, column 10. Every colour holds exactly one queen, so that square is forced. Row 9 and column 10 are now spoken for, along with the squares touching that queen.

  13. Step 13. Amber cannot leave column 11 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 3 squares in all.

  14. Step 14. 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: pale yellow has one square left, so its queen takes row 7, column 9; indigo has one square left, so its queen takes row 1, column 8; amber has one square left, so its queen takes row 2, column 11; mint has one square left, so its queen takes row 4, 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 4, column 12.

    Puzzle #180 after step 14: 7 of 12 queens placed
    The board after step 14 — 7 of 12 queens placed.

  15. Step 15. All of the sea green region's remaining squares (2 open squares) lie in row 10. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.

  16. Step 16. Wherever teal 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.

  17. Step 17. Here is the pairing argument. The red and sea green regions have no open squares outside columns 2 and 3 — 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: 4 squares come off in one stroke.

  18. Step 18. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: teal has one square left, so its queen takes row 11, column 4; sea green has one square left, so its queen takes row 10, column 2; red has one square left, so its queen takes row 6, column 3; rose has one square left, so its queen takes row 12, column 6; blue has one square left, so its queen takes row 3, column 5. 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 3, column 5.

    Puzzle #180 after step 18: 12 of 12 queens placed
    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 #180

This is puzzle number 180 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.

As with every puzzle in this collection, #180 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.

How the colors divide this board

From smallest to largest, the regions on puzzle #180 are: slate (1 square), mint (3 squares), orange (6 squares), red (7 squares), sea green (8 squares), khaki (9 squares), indigo (10 squares), amber (10 squares), pale yellow (16 squares), blue (23 squares), rose (25 squares), teal (26 squares). An even split would hand every color 12 squares, so the slate region is running 11 squares under that average while the teal region is the roomiest area on the grid with 26 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Slate is working with 1 square while teal has 26 to spare, a gap of 25. Attack the cramped end and the roomy regions tend to resolve themselves.

Shape matters as much as size. The mint region is the most compact color here, filling 100% of the 3×1 box that encloses it, while red is the most scattered at 44% of its 4×4 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.

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.

Slate: 1 square across rows 5–5, columns 7–7. It never leaves row 5, so its queen is locked to that row.

Mint covers 3 squares, spanning 3 rows and 1 column (rows 3–5, columns 12–12). It sits wholly inside column 12, so its queen must be in that column.

Orange occupies 6 squares within rows 7–9, columns 10–12.

Red: 7 squares across rows 4–7, columns 2–5.

Sea green covers 8 squares, spanning 5 rows and 3 columns (rows 8–12, columns 1–3).

Khaki occupies 9 squares within rows 2–8, columns 1–2.

Indigo: 10 squares across rows 1–5, columns 7–10.

Amber covers 10 squares, spanning 5 rows and 4 columns (rows 1–5, columns 9–12).

Pale yellow occupies 16 squares within rows 4–7, columns 4–12.

Blue: 23 squares across rows 1–4, columns 1–9.

Rose covers 25 squares, spanning 4 rows and 9 columns (rows 9–12, columns 4–12).

Teal occupies 26 squares within rows 7–12, columns 2–9.

Where to start on this board

The most constrained color is where a solve should begin, and on #180 that is slate with 1 square. It reaches from row 5 to row 5 and from column 7 to column 7 — 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 slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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.

No region here is pinned to a shallow band of rows, so turn the same idea sideways: khaki spans only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.

The tightest lines on puzzle #180

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 6, 10, 11 and 12 are the narrowest, touched by only 3 colors. Column 1 is the tightest column, with 3 colors.

Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 11 rows. Row 5 sits at the far end with 7 colors crossing it; it will almost certainly be among the last to settle.

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.

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 #180 is rated hard

The hard label comes out of four measurable things about #180: 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 25. 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.

A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.

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.

Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 1 on this board.

Use this as a rough check on your progress: the completed board has 4 queens around the border and 8 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.

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 #180?

Puzzle #180 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 #180 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #180, 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 #180?

Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.