7x7 Queens Puzzle #42

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 #42

What follows is a complete solve of puzzle #42, 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: region confined to one line, last square in a region.

Show the full walkthrough (3 steps) — contains the solution

  1. Step 1. Blue cannot leave column 1 — every one of its open squares (3 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 5 squares in all.

  2. Step 2. All of the red region's remaining squares (2 open squares) lie in row 6. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.

    Puzzle #42 after step 2: 0 of 7 queens placed
    The board after step 2 — 0 of 7 queens placed.

  3. Step 3. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: pale yellow has one square left, so its queen takes row 7, column 4; red has one square left, so its queen takes row 6, column 6; mint has one square left, so its queen takes row 4, column 7; indigo has one square left, so its queen takes row 1, column 5; amber has one square left, so its queen takes row 2, column 3; blue has one square left, so its queen takes row 3, column 1; slate has one square left, so its queen takes row 5, 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 5, column 2.

    Puzzle #42 after step 3: 7 of 7 queens placed
    The board after step 3 — 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 #42

Board #42 of the 7x7 bank carries a medium rating. Its 49 squares are shared between seven colored regions, and the solution places seven queens: one per row, one per column, one per region, none of them side by side or corner to corner.

Puzzle #42 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 #42 are: red (2 squares), blue (3 squares), pale yellow (4 squares), mint (5 squares), indigo (7 squares), amber (12 squares), slate (16 squares). An even split would hand every color 7 squares, so the red region is running 5 squares under that average while the slate region is the roomiest area on the grid with 16 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Red is working with 2 squares while slate has 16 to spare, a gap of 14. Attack the cramped end and the roomy regions tend to resolve themselves.

Square count is only half the story — how a color is arranged counts too. Blue is the tidiest shape on #42, packing 100% of its 3×1 bounding box, whereas indigo straggles across its 3×4 box at just 58%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Here is the full footprint of every color on puzzle #42, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Red: 2 squares across rows 6–6, columns 5–6. It never leaves row 6, so its queen is locked to that row.

Blue covers 3 squares, spanning 3 rows and 1 column (rows 1–3, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Pale yellow occupies 4 squares within rows 6–7, columns 4–6.

Mint: 5 squares across rows 4–7, columns 6–7.

Indigo covers 7 squares, spanning 3 rows and 4 columns (rows 1–3, columns 4–7).

Amber occupies 12 squares within rows 1–5, columns 3–6.

Slate: 16 squares across rows 1–7, columns 1–3.

Where to start on this board

Begin where the choices are fewest. On this board that means red, which owns just 2 squares inside a 1×2 window running from row 6 to row 6 and column 5 to column 6. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

Because red never leaves row 6, its queen must be in that row — and that instantly settles row 6 for everyone else. Sweep along it and X out every square that is not red. 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. Pale yellow 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 #42

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 4, 5 and 7 are the narrowest, touched by only 3 colors. Column 2 is the tightest column, with 1 color.

Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 6 rows. Row 6 sits at the far end with 4 colors crossing it; it will almost certainly be among the last to settle.

Working through the middle game

Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.

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.

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 #42 is rated medium

The rating reflects how much the board concedes up front. Puzzle #42 has two regions confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 14 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

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 2 on this board.

Use this as a rough check on your progress: the completed board has 4 queens around the border and 3 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

  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 #42?

Puzzle #42 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.

Does 7x7 puzzle #42 have only one solution?

Yes. Every puzzle in the 7x7 collection, including #42, 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 #42?

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