7x7 Queens Puzzle #127

Difficulty: hard. 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 10-step reasoning is written out below.

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Step-by-step solution for puzzle #127

What follows is a complete solve of puzzle #127, worked the way a person works it rather than the way a computer does. It runs to 10 deductions — 7 queen placements and 3 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 confined to one line, last square in a region, region shadow.

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

  1. Step 1. 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: 7 squares in all.

  2. Step 2. The amber region is down to one open square, at row 2, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 7, the remaining amber squares, and the ring of neighbours around row 2, column 7.

  3. Step 3. Only row 1, column 5 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 5, in indigo and in the ring around that square is now dead.

    Puzzle #127 after step 3: 2 of 7 queens placed
    The board after step 3 — 2 of 7 queens placed.

  4. Step 4. Wherever blue 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.

  5. Step 5. Mint is down to 5 candidates in rows 3–6, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.

  6. Step 6. 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: red has one square left, so its queen takes row 6, column 1; mint has one square left, so its queen takes row 4, column 2; blue has one square left, so its queen takes row 3, column 4; slate has one square left, so its queen takes row 5, column 6; pale yellow has one square left, so its queen takes row 7, column 3. 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 7, column 3.

    Puzzle #127 after step 6: 7 of 7 queens placed
    The board after step 6 — 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 #127

7x7 puzzle #127 is rated hard. 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.

Puzzle #127 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 #127 are: indigo (2 squares), red (2 squares), amber (3 squares), mint (8 squares), slate (9 squares), blue (11 squares), pale yellow (14 squares). An even split would hand every color 7 squares, so the indigo region is running 5 squares under that average while the pale yellow region is the roomiest area on the grid with 14 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 2 squares while pale yellow has 14 to spare, a gap of 12. Attack the cramped end and the roomy regions tend to resolve themselves.

Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×2 frame; slate is the loosest, occupying only 56% of its 4×4 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

Region by region

Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a hard board.

Indigo covers 2 squares, spanning 1 row and 2 columns (rows 1–1, columns 5–6). It never leaves row 1, so its queen is locked to that row.

Red occupies 2 squares within rows 5–6, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

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

Mint covers 8 squares, spanning 6 rows and 2 columns (rows 1–6, columns 1–2).

Slate occupies 9 squares within rows 3–6, columns 4–7.

Blue: 11 squares across rows 1–4, columns 2–5.

Pale yellow covers 14 squares, spanning 3 rows and 7 columns (rows 5–7, columns 1–7).

Where to start on this board

The most constrained color is where a solve should begin, and on #127 that is indigo with 2 squares. It reaches from row 1 to row 1 and from column 5 to column 6 — 1 row by 2 columns. 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 indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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 red 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 #127

A quick way to find the next move is to count how many different colors touch each line. On this board, row 7 is the narrowest, touched by only 1 color. Column 3 is the tightest column, with 2 colors.

A line with few colors is a line close to resolving. Where only 1 color appear, one of them owns the row outright and the others must fit into the remaining 6 rows. Row 6 is the crowded extreme at 4 colors — worth revisiting later, not now.

Working through the middle game

Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.

With seven queens to seat and seven regions competing over 49 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 7x7 boards quickly and players who stall halfway.

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 #127 is rated hard

The hard label comes out of four measurable things about #127: two of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 1 color, and its largest and smallest regions differ by 12. Boards that score tight on those measures give up their first queen fast.

Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.

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.

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 7 on this board.

A non-spoiling distribution check: solved, #127 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

  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 7x7 Queens puzzle #127?

Puzzle #127 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 1 color, and two of its regions are confined to a single row or column, which gives you a forced move early.

Does 7x7 puzzle #127 have only one solution?

Yes. Every puzzle in the 7x7 collection, including #127, 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 #127?

Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–1 and columns 5–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.