9x9 Queens Puzzle #82

Difficulty: medium. Place 9 queens on this 9x9 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 10-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 10 deductions — 9 queen placements and 1 elimination step 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 shadow, last square in a row.

Show the full walkthrough (5 steps) — contains the solution

  1. Step 1. The board opens by cascading. Each of these placements is forced the moment the one before it is marked, so the whole run is really a single sweep repeated: indigo has one square left, so its queen takes row 1, column 6; amber has one square left, so its queen takes row 2, column 8; mint has one square left, so its queen takes row 4, column 9; pale yellow has one square left, so its queen takes row 7, column 1; orange has one square left, so its queen takes row 9, column 2; blue has one square left, so its queen takes row 3, 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 3, column 4.

  2. Step 2. Wherever khaki ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.

  3. Step 3. Row 8 has a single open square left, at column 5 in the khaki region. Every row needs a queen, so this one is forced. Cross out the whole of row 8 and column 5, the remaining khaki squares, and the ring of neighbours around row 8, column 5.

    Puzzle #82 after step 3: 7 of 9 queens placed
    The board after step 3 — 7 of 9 queens placed.

  4. Step 4. 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.

  5. Step 5. The red region is down to one open square, at row 6, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 3, the remaining red squares, and the ring of neighbours around row 6, column 3.

    Puzzle #82 after step 5: 9 of 9 queens placed
    The board after step 5 — 9 of 9 queens placed.

That completes the board: 9 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 9x9 Queens puzzle #82

Board #82 of the 9x9 bank carries a medium rating. Its 81 squares are shared between nine colored regions, and the solution places nine 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: indigo (1 square), pale yellow (1 square), orange (2 squares), amber (5 squares), mint (5 squares), blue (13 squares), red (15 squares), khaki (15 squares), slate (24 squares). An even split would hand every color 9 squares, so the indigo region is running 8 squares under that average while the slate region is the roomiest area on the grid with 24 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 indigo, with 1 square, is under far more pressure than slate with 24. A 23-square gap like this one rewards players who start where the choices are fewest.

A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and khaki the most diffuse at 54% of its 4×7 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

Region by region

Below is every color on #82 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 6–6). It never leaves row 1, so its queen is locked to that row.

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

Orange: 2 squares across rows 9–9, columns 1–2. It never leaves row 9, so its queen is locked to that row.

Amber covers 5 squares, spanning 2 rows and 3 columns (rows 1–2, columns 7–9).

Mint occupies 5 squares within rows 2–4, columns 7–9.

Blue: 13 squares across rows 1–3, columns 1–5.

Red covers 15 squares, spanning 6 rows and 4 columns (rows 3–8, columns 1–4).

Khaki occupies 15 squares within rows 6–9, columns 3–9.

Slate: 24 squares across rows 2–8, columns 4–9.

Where to start on this board

Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 6 and 6. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #82.

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. Pale yellow, orange and amber 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 5 and 9 are the narrowest, touched by only 2 colors. Columns 2, 3, 5 and 6 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 2 colors, so one of that handful must take the row and the rest are squeezed into the other 8 rows. Row 7 sits at the far end with 4 colors crossing it; it will almost certainly be among the last to settle.

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.

Keep alternating between two views of the board. In the region view you ask which of the nine 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 9x9 board unwinds.

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 #82 is rated medium

What earns a board its rating is the shape of its opening. Here that opening offers five single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 23. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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 9x9

At 9x9 a full re-scan after every move starts to get slow, so work locally instead. Once a queen goes down, check only the regions, rows and columns that placement actually touched — that is where new information appeared. Fall back to a complete sweep only when the local pass turns up nothing, and your solve times will drop noticeably.

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 5.

Use this as a rough check on your progress: the completed board has 4 queens around the border and 5 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 9x9 Queens puzzle #82?

Puzzle #82 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.

Does 9x9 puzzle #82 have only one solution?

Yes. Every puzzle in the 9x9 collection, including #82, has been verified to have exactly one valid arrangement of nine 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 indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.