7x7 Queens Puzzle #85
Difficulty: medium. Place 7 queens on this 7x7 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 9-step reasoning is written out below.
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Step-by-step solution for puzzle #85
What follows is a complete solve of puzzle #85, worked the way a person works it rather than the way a computer does. It runs to 9 deductions — 7 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 confined to one line.
Techniques this board needs: last square in a region, region confined to one line.
Show the full walkthrough (5 steps) — contains the solution
Step 1. Only row 4, column 7 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 7, the rest of the mint region and every square adjacent to it — diagonals included.
Step 2. The blue region is down to one open square, at row 3, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 5, the remaining blue squares, and the ring of neighbours around row 3, column 5.
Step 3. Red cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 5 squares in all.
The board after step 3 — 2 of 7 queens placed. Step 4. All of the amber region's remaining squares (2 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 3 squares come off.
Step 5. 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: slate has one square left, so its queen takes row 5, column 3; red has one square left, so its queen takes row 6, column 1; pale yellow has one square left, so its queen takes row 7, column 4; indigo has one square left, so its queen takes row 1, column 6; amber has one square left, so its queen takes row 2, column 2. 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 2, column 2.
The board after step 5 — 7 of 7 queens placed.
That completes the board: 7 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 7x7 Queens puzzle #85
Puzzle #85 is one of the medium-rated boards in the 7x7 set. Seven colors divide its 49 squares, and you win by placing seven queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
Puzzle #85 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 #85 are: mint (1 square), blue (2 squares), amber (4 squares), red (4 squares), pale yellow (8 squares), slate (10 squares), indigo (20 squares). An even split would hand every color 7 squares, so the mint region is running 6 squares under that average while the indigo region is the roomiest area on the grid with 20 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So mint, with 1 square, is under far more pressure than indigo with 20. A 19-square gap like this one rewards players who start where the choices are fewest.
Shape matters as much as size. The amber region is the most compact color here, filling 100% of the 2×2 box that encloses it, while slate is the most scattered at 50% of its 5×4 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.
Region by region
Below is every color on #85 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.
Mint covers 1 square, spanning 1 row and 1 column (rows 4–4, columns 7–7). It never leaves row 4, so its queen is locked to that row.
Blue occupies 2 squares within rows 3–4, columns 5–5. It sits wholly inside column 5, so its queen must be in that column.
Amber: 4 squares across rows 1–2, columns 1–2.
Red covers 4 squares, spanning 4 rows and 1 column (rows 3–6, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.
Pale yellow occupies 8 squares within rows 5–7, columns 3–6.
Slate: 10 squares across rows 3–7, columns 1–4.
Indigo covers 20 squares, spanning 7 rows and 5 columns (rows 1–7, columns 3–7).
Where to start on this board
Mint is the natural opening on #85: it is the smallest region on the grid at 1 square, boxed into rows 4–4 and columns 7–7. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.
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. Amber and blue 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 #85
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 2 are the narrowest, touched by only 2 colors. Columns 2, 6 and 7 are the tightest columns, 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 6 rows. Row 4 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
The middle of #85 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.
The whole middle game reduces to two sweeps. Sweep the seven colors looking for one with a single square left; sweep the seven rows and seven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 7x7 solve from a stalled one.
A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.
Why puzzle #85 is rated medium
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 2 colors, and a region-size spread of 19. 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.
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 7x7
At 7x7 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.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #85, start hunting for it near row 1.
A non-spoiling distribution check: solved, #85 carries 4 queens on the perimeter against 3 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 7x7 Queens puzzle #85?
Puzzle #85 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 7x7 puzzle #85 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #85, has been verified to have exactly one valid arrangement of seven queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #85?
Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.