7x7 Queens Puzzle #157

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 12-step reasoning is written out below.

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Step-by-step solution for puzzle #157

What follows is a complete solve of puzzle #157, worked the way a person works it rather than the way a computer does. It runs to 12 deductions — 7 queen placements and 5 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, region shadow, last square in a region.

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

  1. Step 1. Blue cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 7 squares in all.

  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: 8 squares come off.

  3. Step 3. Pale yellow cannot leave row 7 — every one of its open squares (2 open squares) sits there. That row belongs to pale yellow, so every square on it owned by another colour is dead: 4 squares in all.

    Puzzle #157 after step 3: 0 of 7 queens placed
    The board after step 3 — 0 of 7 queens placed.

  4. Step 4. 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: 2 squares come off.

  5. Step 5. Mint is down to 2 candidates in rows 3 and 4, columns 3 and 4, and they overlap enough to cover 3 squares 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 7 queens fall out in order without any new idea being needed: slate has one square left, so its queen takes row 5, column 5; red has one square left, so its queen takes row 6, column 7; pale yellow has one square left, so its queen takes row 7, column 4; mint has one square left, so its queen takes row 4, column 3; indigo has one square left, so its queen takes row 1, column 2; blue has one square left, so its queen takes row 3, column 1; amber has one square left, so its queen takes row 2, column 6. 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 2, column 6.

    Puzzle #157 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 #157

Puzzle #157 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.

As with every puzzle in this collection, #157 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.

How the colors divide this board

From smallest to largest, the regions on puzzle #157 are: blue (2 squares), slate (2 squares), pale yellow (4 squares), red (6 squares), indigo (9 squares), amber (12 squares), mint (14 squares). An even split would hand every color 7 squares, so the blue region is running 5 squares under that average while the mint region is the roomiest area on the grid with 14 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 blue, with 2 squares, is under far more pressure than mint with 14. A 12-square gap like this one rewards players who start where the choices are fewest.

Shape matters as much as size. The blue region is the most compact color here, filling 100% of the 2×1 box that encloses it, while mint is the most scattered at 56% of its 5×5 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

Below is every color on #157 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

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

Slate occupies 2 squares within rows 5–5, columns 4–5. It never leaves row 5, so its queen is locked to that row.

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

Red covers 6 squares, spanning 3 rows and 2 columns (rows 5–7, columns 6–7).

Indigo occupies 9 squares within rows 1–4, columns 1–4.

Amber: 12 squares across rows 1–4, columns 4–7.

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

Where to start on this board

Blue is the natural opening on #157: it is the smallest region on the grid at 2 squares, boxed into rows 2–3 and columns 1–1. Since one of those squares must hold a queen, that block of 2 rows and 1 column is the most contested part of the board.

Because blue stays inside column 1, its queen belongs to that column and no other region may use it. Cross out the rest of column 1 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. Slate and pale yellow 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 #157

A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Columns 2, 3, 6 and 7 are the tightest columns, with 2 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 3, meeting 4 colors, is the opposite case — the busiest line on #157 and the last one you should expect to crack.

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.

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.

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 #157 is rated medium

What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 2-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 12. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.

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 #157, start hunting for it near row 1.

Here is a shape check that gives nothing away: when #157 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 #157?

Puzzle #157 is rated medium. Its smallest color region holds 2 squares, 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 #157 have only one solution?

Yes. Every puzzle in the 7x7 collection, including #157, 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 #157?

Start with the blue region — the most constrained color on this board, with 2 squares spanning rows 2–3 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.