12x12 Queens Puzzle #82

Difficulty: medium. 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 16-step reasoning is written out below.

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Step-by-step solution for puzzle #82

What follows is a complete solve of puzzle #82, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 12 queen placements and 4 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 region shadow.

Techniques this board needs: last square in a region, region confined to one line, region shadow.

Show the full walkthrough (9 steps) — contains the solution

  1. Step 1. Only row 3, column 12 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 12, the rest of the blue region and every square adjacent to it — diagonals included.

  2. Step 2. The mint region is down to one open square, at row 4, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 5, the remaining mint squares, and the ring of neighbours around row 4, column 5.

  3. Step 3. 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.

    Puzzle #82 after step 3: 3 of 12 queens placed
    The board after step 3 — 3 of 12 queens placed.

  4. Step 4. All of the indigo region's remaining squares (4 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.

  5. Step 5. 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: amber has one square left, so its queen takes row 2, column 2; slate has one square left, so its queen takes row 5, column 8; pale yellow has one square left, so its queen takes row 7, column 11; orange has one square left, so its queen takes row 9, column 9; indigo has one square left, so its queen takes row 1, column 10. 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 1, column 10.

  6. Step 6. Wherever sea green 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.

    Puzzle #82 after step 6: 8 of 12 queens placed
    The board after step 6 — 8 of 12 queens placed.

  7. Step 7. Teal is down to 4 candidates in rows 10–12, columns 3 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about teal itself.

  8. Step 8. All of the sea green region's remaining squares (2 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.

  9. Step 9. 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: rose has one square left, so its queen takes row 12, column 6; sea green has one square left, so its queen takes row 10, column 7; teal has one square left, so its queen takes row 11, column 3; red has one square left, so its queen takes row 6, column 4. 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 6, column 4.

    Puzzle #82 after step 9: 12 of 12 queens placed
    The board after step 9 — 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 #82

Board #82 of the 12x12 bank carries a medium rating. Its 144 squares are shared between twelve colored regions, and the solution places twelve queens: one per row, one per column, one per region, none of them side by side or corner to corner.

As with every puzzle in this collection, #82 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 #82 are: blue (1 square), khaki (1 square), mint (3 squares), amber (7 squares), indigo (8 squares), pale yellow (8 squares), rose (8 squares), teal (13 squares), sea green (18 squares), slate (20 squares), orange (20 squares), red (37 squares). An even split would hand every color 12 squares, so the blue region is running 11 squares under that average while the red region is the roomiest area on the grid with 37 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 red with 37. A 36-square gap like this one rewards players who start where the choices are fewest.

Look at arrangement as well as area. Blue is the densest region on this grid at 100% of a 1×1 frame; sea green is the loosest, occupying only 43% of its 6×7 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

Region by region

Here is the full footprint of every color on puzzle #82, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Blue covers 1 square, spanning 1 row and 1 column (rows 3–3, columns 12–12). It never leaves row 3, so its queen is locked to that row.

Khaki occupies 1 square within rows 8–8, columns 1–1. It never leaves row 8, so its queen is locked to that row.

Mint: 3 squares across rows 3–4, columns 4–5.

Amber covers 7 squares, spanning 3 rows and 4 columns (rows 1–3, columns 1–4).

Indigo occupies 8 squares within rows 1–3, columns 8–12.

Pale yellow: 8 squares across rows 4–8, columns 11–12.

Rose covers 8 squares, spanning 3 rows and 5 columns (rows 10–12, columns 6–10).

Teal occupies 13 squares within rows 9–12, columns 1–4.

Sea green: 18 squares across rows 7–12, columns 4–10.

Slate covers 20 squares, spanning 5 rows and 6 columns (rows 1–5, columns 5–10).

Orange occupies 20 squares within rows 5–12, columns 8–12.

Red: 37 squares across rows 2–10, columns 1–7.

Where to start on this board

Blue is the natural opening on #82: it is the smallest region on the grid at 1 square, boxed into rows 3–3 and columns 12–12. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.

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. Khaki and mint 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 #82

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 6 are the narrowest, touched by only 3 colors. Columns 2, 3 and 11 are the tightest columns, with 3 colors.

A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 11 rows. Row 3 is the crowded extreme at 6 colors — worth revisiting later, not now.

Working through the middle game

After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.

Keep alternating between two views of the board. In the region view you ask which of the twelve colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 12x12 board unwinds.

Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.

Why puzzle #82 is rated medium

The rating reflects how much the board concedes up front. Puzzle #82 has four 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 36 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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 medium 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.

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 #82, start hunting for it near row 1.

A non-spoiling distribution check: solved, #82 carries 4 queens on the perimeter against 8 queens inside, with the corners empty. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.

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 #82?

Puzzle #82 is rated medium. 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 #82 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #82, 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 #82?

Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.