8x8 Queens Puzzle #87
Difficulty: medium. 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 17-step reasoning is written out below.
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Step-by-step solution for puzzle #87
What follows is a complete solve of puzzle #87, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 8 queen placements and 9 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 rows.
Techniques this board needs: region confined to one line, region shadow, last square in a region, pairing argument on rows.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Blue cannot leave column 3 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 10 squares in all.
Step 2. Red is down to 7 candidates in rows 4–8, columns 5–7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 3. Khaki cannot leave column 6 — every one of its open squares (2 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 10 squares in all.
Step 4. Wherever red ends up among its 2 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
The board after step 4 — 0 of 8 queens placed. Step 5. Mint cannot leave column 8 — every one of its open squares (4 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 3 squares in all.
Step 6. All of the amber region's remaining squares (4 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.
Step 7. Only row 6, column 5 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 5, in red and in the ring around that square is now dead.
Step 8. The khaki region is down to one open square, at row 8, column 6. Every colour holds exactly one queen, so that square is forced. Row 8 and column 6 are now spoken for, along with the squares touching that queen.
Step 9. Only row 7, column 1 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 1, the rest of the pale yellow region and every square adjacent to it — diagonals included.
The board after step 9 — 3 of 8 queens placed. Step 10. All of the mint region's remaining squares (2 open squares) lie in column 8. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
Step 11. Slate cannot leave column 2 — every one of its open squares (2 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 12. Here is the pairing argument. The mint and slate regions have no open squares outside rows 4 and 5 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 1 square come off in one stroke.
Step 13. 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: indigo has one square left, so its queen takes row 1, column 4; blue has one square left, so its queen takes row 3, column 3; amber has one square left, so its queen takes row 2, column 7; slate has one square left, so its queen takes row 5, column 2; mint has one square left, so its queen takes row 4, 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 4, column 8.
The board after step 13 — 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 #87
8x8 puzzle #87 is rated medium. 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.
Puzzle #87 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 #87 are: blue (2 squares), khaki (2 squares), pale yellow (6 squares), red (7 squares), mint (8 squares), amber (9 squares), indigo (13 squares), slate (17 squares). An even split would hand every color 8 squares, so the blue region is running 6 squares under that average while the slate region is the roomiest area on the grid with 17 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Blue is working with 2 squares while slate has 17 to spare, a gap of 15. 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 blue is the most solid, filling 100% of its 2×1 box, and red the most diffuse at 47% of its 5×3 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #87 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: 2 squares across rows 2–3, columns 3–3. It sits wholly inside column 3, so its queen must be in that column.
Khaki covers 2 squares, spanning 2 rows and 1 column (rows 7–8, columns 6–6). It sits wholly inside column 6, so its queen must be in that column.
Pale yellow occupies 6 squares within rows 6–8, columns 1–4.
Red: 7 squares across rows 4–8, columns 5–7.
Mint covers 8 squares, spanning 5 rows and 2 columns (rows 4–8, columns 7–8).
Amber occupies 9 squares within rows 1–4, columns 6–8.
Indigo: 13 squares across rows 1–4, columns 1–6.
Slate covers 17 squares, spanning 6 rows and 5 columns (rows 2–7, columns 1–5).
Where to start on this board
Begin where the choices are fewest. On this board that means blue, which owns just 2 squares inside a 2×1 window running from row 2 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 stays inside column 3, its queen belongs to that column and no other region may use it. Cross out the rest of column 3 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. Khaki is no more than two rows tall, so its queen is 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 #87
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 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 7 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
The middle of #87 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.
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 #87 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 2-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 15. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
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 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 1 on this board.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 4 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #87?
Puzzle #87 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 8x8 puzzle #87 have only one solution?
Yes. Every puzzle in the 8x8 collection, including #87, 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 #87?
Start with the blue region — the most constrained color on this board, with 2 squares spanning rows 2–3 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.