9x9 Queens Puzzle #71

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 19-step reasoning is written out below.

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Step-by-step solution for puzzle #71

What follows is a complete solve of puzzle #71, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 9 queen placements and 10 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, pairing argument on rows, line confined to one region, pairing argument on columns, region confined to one line, last square in a column, last square in a region, last square in a row.

Show the full walkthrough (11 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. Slate is down to 3 candidates in rows 5 and 6, columns 5 and 6, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about slate itself.

  3. Step 3. The pale yellow region has 4 squares left, spread across rows 7–9, columns 8 and 9. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.

  4. Step 4. Wherever orange ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.

    Puzzle #71 after step 4: 0 of 9 queens placed
    The board after step 4 — 0 of 9 queens placed.

  5. Step 5. Pale yellow is down to 3 candidates in rows 7–9, columns 8 and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.

  6. Step 6. Here is the pairing argument. The pale yellow, khaki, and orange regions have no open squares outside rows 7–9 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 7 squares come off in one stroke.

  7. Step 7. Column 1 has only blue squares left (6 open squares). The column needs a queen and only blue can provide one, which pins that colour to this column and kills its squares elsewhere: 7 squares.

    Puzzle #71 after step 7: 0 of 9 queens placed
    The board after step 7 — 0 of 9 queens placed.

  8. Step 8. Here is the pairing argument. The indigo and pale yellow regions have no open squares outside columns 8 and 9 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 7 squares come off in one stroke.

  9. Step 9. Mint cannot leave column 7 — every one of its open squares (3 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 5 squares in all.

  10. Step 10. All of the orange region's remaining squares (2 open squares) lie in row 9. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.

  11. Step 11. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: column 2 is down to one square, putting a queen on row 6, column 2; slate has one square left, so its queen takes row 5, column 5; mint has one square left, so its queen takes row 4, column 7; orange has one square left, so its queen takes row 9, column 6; row 3 is down to one square, putting a queen on row 3, column 1; column 9 is down to one square, putting a queen on row 1, column 9; amber has one square left, so its queen takes row 2, column 4; khaki has one square left, so its queen takes row 8, column 3; pale yellow has one square left, so its queen takes row 7, column 8. 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 7, column 8.

    Puzzle #71 after step 11: 9 of 9 queens placed
    The board after step 11 — 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 #71

9x9 puzzle #71 is rated medium. Nine colored regions cover its 81 squares, and your job is to seat nine queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

Puzzle #71 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.

How the colors divide this board

From smallest to largest, the regions on puzzle #71 are: amber (3 squares), slate (3 squares), pale yellow (4 squares), orange (4 squares), indigo (5 squares), khaki (10 squares), blue (14 squares), mint (14 squares), red (24 squares). An even split would hand every color 9 squares, so the amber region is running 6 squares under that average while the red region is the roomiest area on the grid with 24 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 3 squares while red has 24 to spare, a gap of 21. Attack the cramped end and the roomy regions tend to resolve themselves.

Square count is only half the story — how a color is arranged counts too. Blue is the tidiest shape on #71, packing 78% of its 6×3 bounding box, whereas red straggles across its 9×7 box at just 38%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Here is the full footprint of every color on puzzle #71, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Amber occupies 3 squares within rows 1–2, columns 3–4.

Slate: 3 squares across rows 5–6, columns 5–6.

Pale yellow covers 4 squares, spanning 3 rows and 2 columns (rows 7–9, columns 8–9).

Orange occupies 4 squares within rows 8–9, columns 5–7.

Indigo: 5 squares across rows 1–4, columns 8–9.

Khaki covers 10 squares, spanning 3 rows and 5 columns (rows 7–9, columns 2–6).

Blue occupies 14 squares within rows 1–6, columns 1–3.

Mint: 14 squares across rows 2–8, columns 6–9.

Red covers 24 squares, spanning 9 rows and 7 columns (rows 1–9, columns 1–7).

Where to start on this board

Amber is the natural opening on #71: it is the smallest region on the grid at 3 squares, boxed into rows 1–2 and columns 3–4. Since one of those squares must hold a queen, that block of 2 rows and 2 columns is the most contested part of the board.

Amber spans 2 rows, so it stays open for now — but it is still the shortest list on the board at 3 squares. Work through those squares first and mark the ones that would leave another region homeless. That pruning is what makes the rest of the grid legible.

Work outward to the other flat regions next. Slate and orange 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 #71

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 3, 4, 5, 6, 7 and 9 are the narrowest, touched by only 4 colors. Column 1 is the tightest column, with 2 colors.

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 4 colors has to be claimed by one of them, which pushes every remaining region into the other 8 rows. Row 8, meeting 5 colors, is the opposite case — the busiest line on #71 and the last one you should expect to crack.

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.

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.

Resist the plausible-looking queen. On #71 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.

Why puzzle #71 is rated medium

The rating reflects how much the board concedes up front. Puzzle #71 has no region confined to a single row or column, its smallest color holds 3 squares, its tightest row meets just 4 colors, and the gap between its largest and smallest regions is 21 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.

When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.

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.

Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around column 1 on this board.

Here is a shape check that gives nothing away: when #71 is solved, 3 queens sit on the outer edge of the grid and 6 sit inside it, with 1 queen in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.

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 #71?

Puzzle #71 is rated medium. Its smallest color region holds 3 squares, its tightest row is touched by 4 colors, and no region is confined to a single row or column, so the opening move takes some hunting.

Does 9x9 puzzle #71 have only one solution?

Yes. Every puzzle in the 9x9 collection, including #71, 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 #71?

Start with the amber region — the most constrained color on this board, with 3 squares spanning rows 1–2 and columns 3–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.