8x8 Queens Puzzle #32

Difficulty: hard. 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 12-step reasoning is written out below.

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Step-by-step solution for puzzle #32

What follows is a complete solve of puzzle #32, worked the way a person works it rather than the way a computer does. It runs to 12 deductions — 8 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: region shadow, last square in a region, region confined to one line, last square in a row.

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

  1. Step 1. Wherever indigo 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. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 8; indigo has one square left, so its queen takes row 1, column 6; blue has one square left, so its queen takes row 3, column 3; mint has one square left, so its queen takes row 4, 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 4, column 5.

  3. Step 3. All of the slate region's remaining squares (3 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.

    Puzzle #32 after step 3: 4 of 8 queens placed
    The board after step 3 — 4 of 8 queens placed.

  4. Step 4. Wherever red ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

  5. Step 5. All of the pale yellow region's remaining squares (2 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.

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

    Puzzle #32 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 #32

This is puzzle number 32 in the 8x8 collection, rated hard. The grid holds 64 squares split between eight colored regions, and a finished board carries eight 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 #32, 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 eight regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #32 are: amber (2 squares), indigo (3 squares), khaki (5 squares), pale yellow (6 squares), blue (7 squares), mint (13 squares), slate (14 squares), red (14 squares). An even split would hand every color 8 squares, so the amber region is running 6 squares under that average while the red region is the roomiest area on the grid with 14 squares.

Every color takes one queen and one only, whatever its size — which is why the 12 square difference between amber and red is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.

Square count is only half the story — how a color is arranged counts too. Amber is the tidiest shape on #32, packing 100% of its 2×1 bounding box, whereas red straggles across its 5×7 box at just 40%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Below is every color on #32 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.

Amber occupies 2 squares within rows 1–2, columns 8–8. It sits wholly inside column 8, so its queen must be in that column.

Indigo: 3 squares across rows 1–2, columns 6–7.

Khaki covers 5 squares, spanning 3 rows and 2 columns (rows 6–8, columns 1–2).

Pale yellow occupies 6 squares within rows 7–8, columns 4–6.

Blue: 7 squares across rows 1–3, columns 1–3.

Mint covers 13 squares, spanning 4 rows and 6 columns (rows 1–4, columns 1–6).

Slate occupies 14 squares within rows 3–8, columns 6–8.

Red: 14 squares across rows 4–8, columns 1–7.

Where to start on this board

The most constrained color is where a solve should begin, and on #32 that is amber with 2 squares. It reaches from row 1 to row 2 and from column 8 to column 8 — 2 rows by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.

Because amber stays inside column 8, its queen belongs to that column and no other region may use it. Cross out the rest of column 8 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.

Work outward to the other flat regions next. Indigo 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 #32

A quick way to find the next move is to count how many different colors touch each line. On this board, row 5 is 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 7 rows. Row 8 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.

Two checks, run alternately, will carry you through the 64 squares of this grid. First: has any of the eight regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 8x8 boards almost always run one of those checks and forget the other.

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 #32 is rated hard

What earns a board its rating is the shape of its opening. Here that opening offers one single-line region, a 2-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 12. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

Stuck on #32 despite the hard 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 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.

There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #32 is row 5.

Use this as a rough check on your progress: the completed board has 3 queens around the border and 5 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 8x8 Queens puzzle #32?

Puzzle #32 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 2 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.

Does 8x8 puzzle #32 have only one solution?

Yes. Every puzzle in the 8x8 collection, including #32, 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 #32?

Start with the amber region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.