12x12 Queens Puzzle #145

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 21-step reasoning is written out below.

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Step-by-step solution for puzzle #145

What follows is a complete solve of puzzle #145, worked the way a person works it rather than the way a computer does. It runs to 21 deductions — 12 queen placements and 9 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: region shadow, last square in a region, region confined to one line, pairing argument on columns, line confined to one region, last square in a row.

Show the full walkthrough (15 steps) — contains the solution

  1. Step 1. Wherever amber 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.

  2. Step 2. Khaki is down to 7 candidates in rows 4–9, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.

  3. Step 3. Only row 10, column 2 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 2, in sea green and in the ring around that square is now dead.

  4. Step 4. All of the khaki region's remaining squares (5 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.

  5. Step 5. Amber and mint are now boxed into columns 11 and 12 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 13 squares at once.

    Puzzle #145 after step 5: 1 of 12 queens placed
    The board after step 5 — 1 of 12 queens placed.

  6. Step 6. The red region has 3 squares left, spread across rows 5 and 6, columns 9 and 10. 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. Red and orange are now boxed into columns 9 and 10 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 10 squares at once.

  8. Step 8. The indigo region is down to one open square, at row 1, column 8. Every colour holds exactly one queen, so that square is forced. Row 1 and column 8 are now spoken for, along with the squares touching that queen.

  9. Step 9. Only row 2, column 11 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 11, the rest of the amber region and every square adjacent to it — diagonals included.

  10. Step 10. The mint region is down to one open square, at row 4, column 12. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 12, the remaining mint squares, and the ring of neighbours around row 4, column 12.

    Puzzle #145 after step 10: 4 of 12 queens placed
    The board after step 10 — 4 of 12 queens placed.

  11. Step 11. Pale yellow cannot leave column 7 — every one of its open squares (2 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 8 squares in all.

  12. Step 12. The teal region is down to one open square, at row 11, column 6. Every colour holds exactly one queen, so that square is forced. Row 11 and column 6 are now spoken for, along with the squares touching that queen.

  13. Step 13. Row 3 has only blue squares left (3 open squares). The row needs a queen and only blue can provide one, which pins that colour to this row and kills its squares elsewhere: 2 squares.

  14. Step 14. Every open square left in row 12 (2 open squares) belongs to the rose region. Since that row must contain a queen, rose is the colour that supplies it — so rose squares anywhere else on the board are out, 6 squares in all.

  15. Step 15. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: row 9 is down to one square, putting a queen on row 9, column 10; red has one square left, so its queen takes row 6, column 9; pale yellow has one square left, so its queen takes row 7, column 7; slate has one square left, so its queen takes row 5, column 4; khaki has one square left, so its queen takes row 8, column 1; rose has one square left, so its queen takes row 12, column 3; 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 #145 after step 15: 12 of 12 queens placed
    The board after step 15 — 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 #145

Puzzle #145 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.

As with every puzzle in this collection, #145 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 #145 are: sea green (2 squares), amber (3 squares), mint (6 squares), red (6 squares), khaki (7 squares), indigo (8 squares), pale yellow (10 squares), slate (13 squares), teal (15 squares), orange (21 squares), rose (22 squares), blue (31 squares). An even split would hand every color 12 squares, so the sea green region is running 10 squares under that average while the blue region is the roomiest area on the grid with 31 squares.

Every color takes one queen and one only, whatever its size — which is why the 29 square difference between sea green and blue is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.

Square count is only half the story — how a color is arranged counts too. Sea green is the tidiest shape on #145, packing 100% of its 1×2 bounding box, whereas pale yellow straggles across its 6×3 box at just 56%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

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.

Sea green covers 2 squares, spanning 1 row and 2 columns (rows 10–10, columns 1–2). It never leaves row 10, so its queen is locked to that row.

Amber occupies 3 squares within rows 1–2, columns 11–12.

Mint: 6 squares across rows 2–5, columns 11–12.

Red covers 6 squares, spanning 2 rows and 4 columns (rows 5–6, columns 9–12).

Khaki occupies 7 squares within rows 4–9, columns 1–2.

Indigo: 8 squares across rows 1–4, columns 8–10.

Pale yellow covers 10 squares, spanning 6 rows and 3 columns (rows 2–7, columns 7–9).

Slate occupies 13 squares within rows 5–8, columns 2–6.

Teal: 15 squares across rows 8–12, columns 6–10.

Orange covers 21 squares, spanning 6 rows and 4 columns (rows 7–12, columns 9–12).

Rose occupies 22 squares within rows 7–12, columns 1–6.

Blue: 31 squares across rows 1–5, columns 1–7.

Where to start on this board

Open on the tightest color. Here that is sea green, confined to 2 squares between rows 10 and 10 and columns 1 and 2. Its queen has nowhere else to go, which means those 1 row and 2 columns carry more constraint than anywhere else on #145.

Because sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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 red 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 #145

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 11 and 12 are the narrowest, touched by only 3 colors. Columns 3, 4, 5, 7 and 8 are the tightest columns, 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 6 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.

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 #145 is rated hard

The hard label comes out of four measurable things about #145: one of its regions is locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 29. Boards that score tight on those measures give up their first queen fast.

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.

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.

There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #145 is row 1.

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 #145?

Puzzle #145 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.

Does 12x12 puzzle #145 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #145, 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 #145?

Start with the sea green region — the most constrained color on this board, with 2 squares spanning rows 10–10 and columns 1–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.