7x7 Queens Puzzle #51
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 #51
What follows is a complete solve of puzzle #51, 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 5, column 4 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 4, the rest of the slate region and every square adjacent to it — diagonals included.
Step 2. The pale yellow region is down to one open square, at row 7, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 1, the remaining pale yellow squares, and the ring of neighbours around row 7, column 1.
Step 3. Indigo cannot leave row 1 — every one of its open squares (3 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 3 squares in all.
The board after step 3 — 2 of 7 queens placed. Step 4. All of the amber region's remaining squares (3 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 1 square 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: blue has one square left, so its queen takes row 3, column 6; mint has one square left, so its queen takes row 4, column 2; amber has one square left, so its queen takes row 2, column 3; red has one square left, so its queen takes row 6, column 7; indigo has one square left, so its queen takes row 1, 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 1, column 5.
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 #51
7x7 puzzle #51 is rated medium. Seven colored regions cover its 49 squares, and your job is to seat seven queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
As with every puzzle in this collection, #51 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.
How the colors divide this board
From smallest to largest, the regions on puzzle #51 are: slate (1 square), pale yellow (1 square), blue (3 squares), indigo (4 squares), amber (5 squares), red (11 squares), mint (24 squares). An even split would hand every color 7 squares, so the slate region is running 6 squares under that average while the mint region is the roomiest area on the grid with 24 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Slate is working with 1 square while mint has 24 to spare, a gap of 23. 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 indigo is the most solid, filling 100% of its 1×4 box, and mint the most diffuse at 57% of its 7×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #51 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.
Slate: 1 square across rows 5–5, columns 4–4. It never leaves row 5, so its queen is locked to that row.
Pale yellow covers 1 square, spanning 1 row and 1 column (rows 7–7, columns 1–1). It never leaves row 7, so its queen is locked to that row.
Blue occupies 3 squares within rows 2–3, columns 6–7.
Indigo: 4 squares across rows 1–1, columns 4–7. It never leaves row 1, so its queen is locked to that row.
Amber covers 5 squares, spanning 2 rows and 4 columns (rows 1–2, columns 2–5).
Red occupies 11 squares within rows 3–7, columns 5–7.
Mint: 24 squares across rows 1–7, columns 1–6.
Where to start on this board
The most constrained color is where a solve should begin, and on #51 that is slate with 1 square. It reaches from row 5 to row 5 and from column 4 to column 4 — 1 row 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 slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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. Indigo, pale yellow, 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 #51
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 4 and 6 are the narrowest, touched by only 2 colors. Columns 1, 2 and 3 are the tightest columns, with 2 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 6 rows. Row 7, meeting 3 colors, is the opposite case — the busiest line on #51 and the last one you should expect to crack.
Working through the middle game
The middle of #51 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.
Keep alternating between two views of the board. In the region view you ask which of the seven colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 7x7 board unwinds.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #51 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers five single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 23. 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.
A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.
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.
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 4 on this board.
Here is a shape check that gives nothing away: when #51 is solved, 3 queens sit on the outer edge of the grid and 4 sit inside it, with 1 queen in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 #51?
Puzzle #51 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.
Does 7x7 puzzle #51 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #51, 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 #51?
Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.