7x7 Queens Puzzle #68

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 7-step reasoning is written out below.

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Step-by-step solution for puzzle #68

What follows is a complete solve of puzzle #68, worked the way a person works it rather than the way a computer does. It runs to 7 deductions, every one of them a forced placement. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the last square in a row.

Techniques this board needs: last square in a region, last square in a row.

Show the full walkthrough (7 steps) — contains the solution

  1. Step 1. Only row 4, column 1 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 1, the rest of the mint region and every square adjacent to it — diagonals included.

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

  3. Step 3. Only row 5, column 6 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 6, in slate and in the ring around that square is now dead.

  4. Step 4. The blue region is down to one open square, at row 3, column 4. Every colour holds exactly one queen, so that square is forced. Row 3 and column 4 are now spoken for, along with the squares touching that queen.

    Puzzle #68 after step 4: 4 of 7 queens placed
    The board after step 4 — 4 of 7 queens placed.

  5. Step 5. Look along row 2: one square survives, row 2, column 2, which belongs to amber. A row cannot be left empty, so the queen goes there. Everything else in row 2, in column 2, in amber and in the ring around that square is now dead.

  6. Step 6. The indigo region is down to one open square, at row 1, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 5, the remaining indigo squares, and the ring of neighbours around row 1, column 5.

  7. Step 7. Only row 6, column 3 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 3, in red and in the ring around that square is now dead.

    Puzzle #68 after step 7: 7 of 7 queens placed
    The board after step 7 — 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 #68

This is puzzle number 68 in the 7x7 collection, rated medium. The grid holds 49 squares split between seven colored regions, and a finished board carries seven queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

Puzzle #68 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 #68 are: mint (1 square), pale yellow (1 square), blue (4 squares), slate (8 squares), amber (10 squares), indigo (11 squares), red (14 squares). An even split would hand every color 7 squares, so the mint region is running 6 squares under that average while the red region is the roomiest area on the grid with 14 squares.

Every color takes one queen and one only, whatever its size — which is why the 13 square difference between mint and red is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.

Shape matters as much as size. The mint region is the most compact color here, filling 100% of the 1×1 box that encloses it, while blue is the most scattered at 67% of its 3×2 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

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 medium board.

Mint occupies 1 square within rows 4–4, columns 1–1. It never leaves row 4, so its queen is locked to that row.

Pale yellow: 1 square across rows 7–7, columns 7–7. It never leaves row 7, so its queen is locked to that row.

Blue covers 4 squares, spanning 3 rows and 2 columns (rows 3–5, columns 4–5).

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

Amber: 10 squares across rows 1–5, columns 1–3.

Indigo covers 11 squares, spanning 3 rows and 5 columns (rows 1–3, columns 3–7).

Red occupies 14 squares within rows 4–7, columns 1–5.

Where to start on this board

Open on the tightest color. Here that is mint, confined to 1 square between rows 4 and 4 and columns 1 and 1. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #68.

Because mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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 #68

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2 and 6 are the narrowest, touched by only 2 colors. Columns 2 and 6 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 4 sits at the far end with 5 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.

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.

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 #68 is rated medium

The medium label comes out of four measurable things about #68: four of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 13. Boards that score tight on those measures give up their first queen fast.

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.

There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #68 is row 1.

Use this as a rough check on your progress: the completed board has 3 queens around the border and 4 queens in the middle, including 1 in a corner. 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 #68?

Puzzle #68 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.

Does 7x7 puzzle #68 have only one solution?

Yes. Every puzzle in the 7x7 collection, including #68, 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 #68?

Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.