9x9 Queens Puzzle #52

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 9-step reasoning is written out below.

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Step-by-step solution for puzzle #52

What follows is a complete solve of puzzle #52, worked the way a person works it rather than the way a computer does. It runs to 9 deductions, every one of them a forced placement. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the last square in a region.

Techniques this board needs: last square in a region.

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

  1. Step 1. Only row 9, column 4 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 4, the rest of the orange 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 3, column 2 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 2, in blue and in the ring around that square is now dead.

    Puzzle #52 after step 3: 3 of 9 queens placed
    The board after step 3 — 3 of 9 queens placed.

  4. Step 4. The indigo region is down to one open square, at row 1, column 1. Every colour holds exactly one queen, so that square is forced. Row 1 and column 1 are now spoken for, along with the squares touching that queen.

  5. Step 5. Only row 6, column 6 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 6, the rest of the red region and every square adjacent to it — diagonals included.

  6. Step 6. The pale yellow region is down to one open square, at row 7, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 9, the remaining pale yellow squares, and the ring of neighbours around row 7, column 9.

    Puzzle #52 after step 6: 6 of 9 queens placed
    The board after step 6 — 6 of 9 queens placed.

  7. Step 7. Only row 5, column 3 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 3, in slate and in the ring around that square is now dead.

  8. Step 8. The khaki region is down to one open square, at row 8, column 7. Every colour holds exactly one queen, so that square is forced. Row 8 and column 7 are now spoken for, along with the squares touching that queen.

  9. Step 9. Only row 2, column 8 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 8, the rest of the amber region and every square adjacent to it — diagonals included.

    Puzzle #52 after step 9: 9 of 9 queens placed
    The board after step 9 — 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 #52

This is puzzle number 52 in the 9x9 collection, rated medium. The grid holds 81 squares split between nine colored regions, and a finished board carries nine queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

There is exactly one way to finish #52, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these nine regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #52 are: orange (1 square), pale yellow (2 squares), blue (3 squares), mint (3 squares), red (3 squares), indigo (5 squares), khaki (17 squares), amber (20 squares), slate (27 squares). An even split would hand every color 9 squares, so the orange region is running 8 squares under that average while the slate region is the roomiest area on the grid with 27 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 orange, with 1 square, is under far more pressure than slate with 27. A 26-square gap like this one rewards players who start where the choices are fewest.

Look at arrangement as well as area. Pale yellow is the densest region on this grid at 100% of a 2×1 frame; slate is the loosest, occupying only 50% of its 9×6 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 #52, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Orange covers 1 square, spanning 1 row and 1 column (rows 9–9, columns 4–4). It never leaves row 9, so its queen is locked to that row.

Pale yellow occupies 2 squares within rows 6–7, columns 9–9. It sits wholly inside column 9, so its queen must be in that column.

Blue: 3 squares across rows 3–4, columns 1–2.

Mint covers 3 squares, spanning 2 rows and 2 columns (rows 3–4, columns 4–5).

Red occupies 3 squares within rows 6–7, columns 5–6.

Indigo: 5 squares across rows 1–3, columns 1–2.

Khaki covers 17 squares, spanning 5 rows and 5 columns (rows 5–9, columns 5–9).

Amber occupies 20 squares within rows 1–5, columns 5–9.

Slate: 27 squares across rows 1–9, columns 1–6.

Where to start on this board

Begin where the choices are fewest. On this board that means orange, which owns just 1 square inside a 1×1 window running from row 9 to row 9 and column 4 to column 4. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

Because orange never leaves row 9, its queen must be in that row — and that instantly settles row 9 for everyone else. Sweep along it and X out every square that is not orange. 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. Blue, mint, red and pale yellow 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 #52

A quick way to find the next move is to count how many different colors touch each line. On this board, row 8 is the narrowest, touched by only 2 colors. Column 3 is the tightest column, with 1 color.

A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 8 rows. Row 3 is the crowded extreme at 5 colors — worth revisiting later, not now.

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.

With nine queens to seat and nine regions competing over 81 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 9x9 boards quickly and players who stall halfway.

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 #52 is rated medium

The rating reflects how much the board concedes up front. Puzzle #52 has three 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 26 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.

Stuck on #52 despite the medium rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.

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 #52, start hunting for it near column 3.

A non-spoiling distribution check: solved, #52 carries 3 queens on the perimeter against 6 queens inside, 1 of them cornered. 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 9x9 Queens puzzle #52?

Puzzle #52 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.

Does 9x9 puzzle #52 have only one solution?

Yes. Every puzzle in the 9x9 collection, including #52, 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 #52?

Start with the orange region — the most constrained color on this board, with 1 square spanning rows 9–9 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.