7x7 Queens Puzzle #181

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 8-step reasoning is written out below.

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Step-by-step solution for puzzle #181

What follows is a complete solve of puzzle #181, worked the way a person works it rather than the way a computer does. It runs to 8 deductions — 7 queen placements and 1 elimination step 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 (3 steps) — contains the solution

  1. Step 1. Only row 7, column 6 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 6, the rest of the pale yellow region and every square adjacent to it — diagonals included.

  2. Step 2. All of the slate region's remaining squares (2 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 7 squares come off.

    Puzzle #181 after step 2: 1 of 7 queens placed
    The board after step 2 — 1 of 7 queens placed.

  3. Step 3. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 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 3; amber has one square left, so its queen takes row 2, column 2; blue has one square left, so its queen takes row 3, column 7; slate has one square left, so its queen takes row 5, column 5; indigo has one square left, so its queen takes row 1, column 4. 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 4.

    Puzzle #181 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 #181

Puzzle #181 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.

There is exactly one way to finish #181, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these seven regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #181 are: pale yellow (1 square), red (5 squares), mint (6 squares), blue (8 squares), slate (8 squares), amber (10 squares), indigo (11 squares). An even split would hand every color 7 squares, so the pale yellow region is running 6 squares under that average while the indigo region is the roomiest area on the grid with 11 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 pale yellow, with 1 square, is under far more pressure than indigo with 11. A 10-square gap like this one rewards players who start where the choices are fewest.

Shape matters as much as size. The pale yellow region is the most compact color here, filling 100% of the 1×1 box that encloses it, while blue is the most scattered at 53% of its 5×3 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

Here is the full footprint of every color on puzzle #181, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Pale yellow covers 1 square, spanning 1 row and 1 column (rows 7–7, columns 6–6). It never leaves row 7, so its queen is locked to that row.

Red occupies 5 squares within rows 5–7, columns 1–2.

Mint: 6 squares across rows 4–6, columns 2–4.

Blue covers 8 squares, spanning 5 rows and 3 columns (rows 3–7, columns 5–7).

Slate occupies 8 squares within rows 5–7, columns 3–6.

Amber: 10 squares across rows 1–4, columns 1–4.

Indigo covers 11 squares, spanning 3 rows and 6 columns (rows 1–3, columns 2–7).

Where to start on this board

Pale yellow is the natural opening on #181: it is the smallest region on the grid at 1 square, boxed into rows 7–7 and columns 6–6. 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 pale yellow never leaves row 7, its queen must be in that row — and that instantly settles row 7 for everyone else. Sweep along it and X out every square that is not pale yellow. 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.

No region here is pinned to a shallow band of rows, so turn the same idea sideways: red spans only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.

The tightest lines on puzzle #181

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 1 and 7 are the tightest columns, 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 6 rows. Row 7 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.

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 #181 is rated medium

The rating reflects how much the board concedes up front. Puzzle #181 has two regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 10 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

Stuck on #181 despite the medium rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.

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 #181, start hunting for it near row 1.

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 #181?

Puzzle #181 is rated medium. Its smallest color region holds 1 square, 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 7x7 puzzle #181 have only one solution?

Yes. Every puzzle in the 7x7 collection, including #181, 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 #181?

Start with the pale yellow region — the most constrained color on this board, with 1 square spanning rows 7–7 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.