9x9 Queens Puzzle #54

Difficulty: hard. 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 13-step reasoning is written out below.

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Step-by-step solution for puzzle #54

What follows is a complete solve of puzzle #54, worked the way a person works it rather than the way a computer does. It runs to 13 deductions — 9 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 (7 steps) — contains the solution

  1. Step 1. Only row 2, column 6 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 6, the rest of the amber region and every square adjacent to it — diagonals included.

  2. Step 2. The orange region is down to one open square, at row 9, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 9, the remaining orange squares, and the ring of neighbours around row 9, column 9.

  3. Step 3. Indigo cannot leave row 1 — every one of its open squares (4 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 1 square in all.

  4. Step 4. Wherever mint 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.

    Puzzle #54 after step 4: 2 of 9 queens placed
    The board after step 4 — 2 of 9 queens placed.

  5. Step 5. Slate cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 7 squares in all.

  6. Step 6. All of the blue region's remaining squares (2 open squares) lie in row 3. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 3 squares come off.

  7. Step 7. 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: mint has one square left, so its queen takes row 4, column 7; slate has one square left, so its queen takes row 5, column 1; red has one square left, so its queen takes row 6, column 3; blue has one square left, so its queen takes row 3, column 2; indigo has one square left, so its queen takes row 1, column 4; pale yellow has one square left, so its queen takes row 7, column 8; khaki has one square left, so its queen takes row 8, 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 8, column 5.

    Puzzle #54 after step 7: 9 of 9 queens placed
    The board after step 7 — 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 #54

Board #54 of the 9x9 bank carries a hard 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, #54 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 #54 are: amber (1 square), orange (1 square), slate (2 squares), blue (4 squares), indigo (11 squares), red (13 squares), pale yellow (13 squares), mint (16 squares), khaki (20 squares). An even split would hand every color 9 squares, so the amber region is running 8 squares under that average while the khaki region is the roomiest area on the grid with 20 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Amber has only 1 place to put its queen; khaki has 20. The spread between them on this board is 19 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Shape matters as much as size. The amber region is the most compact color here, filling 100% of the 1×1 box that encloses it, while red is the most scattered at 52% of its 5×5 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

Here is the full footprint of every color on puzzle #54, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Amber: 1 square across rows 2–2, columns 6–6. It never leaves row 2, so its queen is locked to that row.

Orange covers 1 square, spanning 1 row and 1 column (rows 9–9, columns 9–9). It never leaves row 9, so its queen is locked to that row.

Slate occupies 2 squares within rows 4–5, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Blue: 4 squares across rows 3–4, columns 1–3.

Indigo covers 11 squares, spanning 2 rows and 6 columns (rows 1–2, columns 1–6).

Red occupies 13 squares within rows 3–7, columns 1–5.

Pale yellow: 13 squares across rows 4–9, columns 7–9.

Mint covers 16 squares, spanning 6 rows and 4 columns (rows 1–6, columns 6–9).

Khaki occupies 20 squares within rows 6–9, columns 1–7.

Where to start on this board

Open on the tightest color. Here that is amber, confined to 1 square between rows 2 and 2 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 #54.

Because amber never leaves row 2, its queen must be in that row — and that instantly settles row 2 for everyone else. Sweep along it and X out every square that is not amber. 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. Orange, indigo, blue and slate 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 #54

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 8 are the narrowest, touched by only 2 colors. Column 8 is the tightest column, with 2 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 4 sits at the far end with 5 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 nine colors looking for one with a single square left; sweep the nine rows and nine columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 9x9 solve from a stalled one.

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 #54 is rated hard

The rating reflects how much the board concedes up front. Puzzle #54 has five regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 19 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

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

Use this as a rough check on your progress: the completed board has 3 queens around the border and 6 queens in the middle, including 1 in a corner. 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 #54?

Puzzle #54 is rated hard. 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 #54 have only one solution?

Yes. Every puzzle in the 9x9 collection, including #54, 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 #54?

Start with the amber region — the most constrained color on this board, with 1 square spanning rows 2–2 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.