7x7 Queens Puzzle #67

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 11-step reasoning is written out below.

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Step-by-step solution for puzzle #67

What follows is a complete solve of puzzle #67, worked the way a person works it rather than the way a computer does. It runs to 11 deductions — 7 queen placements and 4 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 shadow, region confined to one line, last square in a region.

Show the full walkthrough (8 steps) — contains the solution

  1. Step 1. Wherever amber ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.

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

  3. Step 3. Mint cannot leave column 1 — every one of its open squares (3 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 4 squares in all.

  4. Step 4. The pale yellow region is down to one open square, at row 7, column 2. Every colour holds exactly one queen, so that square is forced. Row 7 and column 2 are now spoken for, along with the squares touching that queen.

    Puzzle #67 after step 4: 1 of 7 queens placed
    The board after step 4 — 1 of 7 queens placed.

  5. Step 5. Only row 2, column 3 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 3, the rest of the amber region and every square adjacent to it — diagonals included.

  6. Step 6. The blue region is down to one open square, at row 3, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 7, the remaining blue squares, and the ring of neighbours around row 3, column 7.

  7. Step 7. Wherever slate ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

  8. Step 8. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: red has one square left, so its queen takes row 6, column 6; slate has one square left, so its queen takes row 5, column 4; indigo has one square left, so its queen takes row 1, column 5; mint has one square left, so its queen takes row 4, column 1. 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 4, column 1.

    Puzzle #67 after step 8: 7 of 7 queens placed
    The board after step 8 — 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 #67

7x7 puzzle #67 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.

Puzzle #67 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 #67 are: blue (2 squares), pale yellow (2 squares), mint (3 squares), amber (4 squares), red (7 squares), indigo (11 squares), slate (20 squares). An even split would hand every color 7 squares, so the blue region is running 5 squares under that average while the slate region is the roomiest area on the grid with 20 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Blue has only 2 places to put its queen; slate has 20. The spread between them on this board is 18 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Look at arrangement as well as area. Blue is the densest region on this grid at 100% of a 2×1 frame; indigo is the loosest, occupying only 55% of its 4×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

Region by region

The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.

Blue covers 2 squares, spanning 2 rows and 1 column (rows 2–3, columns 7–7). It sits wholly inside column 7, so its queen must be in that column.

Pale yellow occupies 2 squares within rows 7–7, columns 1–2. It never leaves row 7, so its queen is locked to that row.

Mint: 3 squares across rows 4–6, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Amber covers 4 squares, spanning 2 rows and 3 columns (rows 1–2, columns 1–3).

Red occupies 7 squares within rows 6–7, columns 3–7.

Indigo: 11 squares across rows 1–4, columns 3–7.

Slate covers 20 squares, spanning 5 rows and 7 columns (rows 2–6, columns 1–7).

Where to start on this board

The most constrained color is where a solve should begin, and on #67 that is blue with 2 squares. It reaches from row 2 to row 3 and from column 7 to column 7 — 2 rows 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 blue stays inside column 7, its queen belongs to that column and no other region may use it. Cross out the rest of column 7 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.

Work outward to the other flat regions next. Pale yellow, 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 #67

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 5 and 7 are the narrowest, touched by only 2 colors. Columns 2, 4, 5 and 6 are the tightest columns, with 3 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 2, meeting 4 colors, is the opposite case — the busiest line on #67 and the last one you should expect to crack.

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.

Two checks, run alternately, will carry you through the 49 squares of this grid. First: has any of the seven regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 7x7 boards almost always run one of those checks and forget the other.

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 #67 is rated medium

Ratings track how generous the opening is. On #67 you get three regions pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 2 colors, and a 18-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.

Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.

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.

One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #67 the place to look for one is around row 1.

Here is a shape check that gives nothing away: when #67 is solved, 4 queens sit on the outer edge of the grid and 3 sit inside it, and no queen lands 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

  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 #67?

Puzzle #67 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.

Does 7x7 puzzle #67 have only one solution?

Yes. Every puzzle in the 7x7 collection, including #67, 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 #67?

Start with the blue region — the most constrained color on this board, with 2 squares spanning rows 2–3 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.