8x8 Queens Puzzle #31

Difficulty: easy. Place 8 queens on this 8x8 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 #31

What follows is a complete solve of puzzle #31, worked the way a person works it rather than the way a computer does. It runs to 10 deductions — 8 queen placements and 2 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 shadow, region confined to one line.

Show the full walkthrough (6 steps) — contains the solution

  1. Step 1. Only row 4, column 8 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 8, the rest of the mint region and every square adjacent to it — diagonals included.

  2. Step 2. The indigo region is down to one open square, at row 1, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 7, the remaining indigo squares, and the ring of neighbours around row 1, column 7.

  3. Step 3. Only row 5, column 1 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 1, in slate and in the ring around that square is now dead.

    Puzzle #31 after step 3: 3 of 8 queens placed
    The board after step 3 — 3 of 8 queens placed.

  4. Step 4. 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.

  5. Step 5. Red cannot leave row 6 — every one of its open squares (2 open squares) sits there. That row belongs to red, so every square on it owned by another colour is dead: 4 squares in all.

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

    Puzzle #31 after step 6: 8 of 8 queens placed
    The board after step 6 — 8 of 8 queens placed.

That completes the board: 8 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 8x8 Queens puzzle #31

8x8 puzzle #31 is rated easy. Eight colored regions cover its 64 squares, and your job is to seat eight queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

Like every board on this site, #31 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.

How the colors divide this board

From smallest to largest, the regions on puzzle #31 are: mint (1 square), slate (2 squares), indigo (4 squares), red (4 squares), amber (6 squares), khaki (8 squares), pale yellow (13 squares), blue (26 squares). An even split would hand every color 8 squares, so the mint region is running 7 squares under that average while the blue region is the roomiest area on the grid with 26 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Mint is working with 1 square while blue has 26 to spare, a gap of 25. Attack the cramped end and the roomy regions tend to resolve themselves.

A region's outline can constrain it more than its size does. Here mint is the most solid, filling 100% of its 1×1 box, and blue the most diffuse at 65% of its 5×8 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

Region by region

The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.

Mint covers 1 square, spanning 1 row and 1 column (rows 4–4, columns 8–8). It never leaves row 4, 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.

Indigo: 4 squares across rows 1–3, columns 7–8.

Red covers 4 squares, spanning 2 rows and 3 columns (rows 5–6, columns 5–7).

Amber occupies 6 squares within rows 1–3, columns 1–3.

Khaki: 8 squares across rows 6–8, columns 5–8.

Pale yellow covers 13 squares, spanning 3 rows and 5 columns (rows 6–8, columns 1–5).

Blue occupies 26 squares within rows 1–5, columns 1–8.

Where to start on this board

Begin where the choices are fewest. On this board that means mint, which owns just 1 square inside a 1×1 window running from row 4 to row 4 and column 8 to column 8. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

Because mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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. Slate and red 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 #31

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 7 and 8 are the narrowest, touched by only 2 colors. Column 4 is the tightest column, with 2 colors.

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 7 rows. Row 6 is the crowded extreme at 3 colors — worth revisiting later, not now.

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 eight queens to seat and eight regions competing over 64 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 8x8 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 #31 is rated easy

Ratings track how generous the opening is. On #31 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 25-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.

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.

If #31 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.

Techniques that work well at 8x8

At 8x8 the whole grid stays within a single glance, which makes systematic scanning practical. Read the rows in order and count the colors in each — a row touched by two colors is far more constrained than one touched by five, and that is usually where the next forced move hides. The board is small enough that re-scanning from the top after each placement costs almost nothing.

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 row 7 on this board.

A non-spoiling distribution check: solved, #31 carries 4 queens on the perimeter against 4 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 8x8 Queens puzzle #31?

Puzzle #31 is rated easy. 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 8x8 puzzle #31 have only one solution?

Yes. Every puzzle in the 8x8 collection, including #31, has been verified to have exactly one valid arrangement of eight queens, and it can always be reached by logic alone — no guessing or backtracking needed.

Where should I place my first queen on puzzle #31?

Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.