8x8 Queens Puzzle #131
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 #131
What follows is a complete solve of puzzle #131, 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: last square in a region, region confined to one line, region shadow.
Show the full walkthrough (9 steps) — contains the solution
Step 1. Only row 3, column 3 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 3, the rest of the blue region and every square adjacent to it — diagonals included.
Step 2. The khaki region is down to one open square, at row 8, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 4, the remaining khaki squares, and the ring of neighbours around row 8, column 4.
Step 3. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 5 squares in all.
The board after step 3 — 2 of 8 queens placed. Step 4. The amber region is down to one open square, at row 2, column 8. Every colour holds exactly one queen, so that square is forced. Row 2 and column 8 are now spoken for, along with the squares touching that queen.
Step 5. Slate cannot leave column 1 — every one of its open squares (4 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 1 square in all.
Step 6. The indigo region is down to one open square, at row 1, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 2, the remaining indigo squares, and the ring of neighbours around row 1, column 2.
The board after step 6 — 4 of 8 queens placed. Step 7. 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.
Step 8. All of the pale yellow region's remaining squares (2 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
Step 9. 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: mint has one square left, so its queen takes row 4, column 6; red has one square left, so its queen takes row 6, column 5; pale yellow has one square left, so its queen takes row 7, column 7; slate has one square left, so its queen takes row 5, column 1. 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 1.
The board after step 9 — 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 #131
8x8 puzzle #131 is rated hard. 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.
There is exactly one way to finish #131, 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 #131 are: blue (1 square), amber (3 squares), khaki (4 squares), indigo (7 squares), slate (7 squares), pale yellow (8 squares), mint (16 squares), red (18 squares). An even split would hand every color 8 squares, so the blue region is running 7 squares under that average while the red region is the roomiest area on the grid with 18 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Blue is working with 1 square while red has 18 to spare, a gap of 17. 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 amber is the most solid, filling 100% of its 3×1 box, and indigo the most diffuse at 58% of its 3×4 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #131 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.
Blue occupies 1 square within rows 3–3, columns 3–3. It never leaves row 3, so its queen is locked to that row.
Amber: 3 squares across rows 1–3, columns 8–8. It sits wholly inside column 8, so its queen must be in that column.
Khaki covers 4 squares, spanning 3 rows and 2 columns (rows 6–8, columns 3–4).
Indigo occupies 7 squares within rows 1–3, columns 1–4.
Slate: 7 squares across rows 2–8, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Pale yellow covers 8 squares, spanning 5 rows and 2 columns (rows 4–8, columns 7–8).
Mint occupies 16 squares within rows 1–6, columns 4–7.
Red: 18 squares across rows 4–8, columns 2–6.
Where to start on this board
Begin where the choices are fewest. On this board that means blue, which owns just 1 square inside a 1×1 window running from row 3 to row 3 and column 3 to column 3. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: pale yellow and khaki span only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #131
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 3 colors. Columns 1, 2, 5, 6, 7 and 8 are the tightest columns, with 2 colors.
A line with few colors is a line close to resolving. Where only 3 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 5 colors — worth revisiting later, not now.
Working through the middle game
The middle of #131 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.
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.
Resist the plausible-looking queen. On #131 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 #131 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers four single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 17. 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 #131 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.
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.
A non-spoiling distribution check: solved, #131 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
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- 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.
- 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 #131?
Puzzle #131 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 8x8 puzzle #131 have only one solution?
Yes. Every puzzle in the 8x8 collection, including #131, 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 #131?
Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.