7x7 Queens Puzzle #176

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 #176

What follows is a complete solve of puzzle #176, 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 confined to one line, region shadow, last square in a region.

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

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

  2. Step 2. Red is down to 5 candidates in rows 6 and 7, columns 4–7, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about red itself.

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

  4. Step 4. The red region is down to one open square, at row 6, column 5. Every colour holds exactly one queen, so that square is forced. Row 6 and column 5 are now spoken for, along with the squares touching that queen.

    Puzzle #176 after step 4: 2 of 7 queens placed
    The board after step 4 — 2 of 7 queens placed.

  5. Step 5. Indigo is down to 3 candidates in rows 1 and 2, columns 3 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.

  6. Step 6. The blue region has 4 squares left, spread across rows 3–5, columns 2 and 3. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.

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

    Puzzle #176 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 #176

This is puzzle number 176 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.

There is exactly one way to finish #176, 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 #176 are: amber (2 squares), pale yellow (2 squares), red (5 squares), indigo (7 squares), blue (7 squares), slate (11 squares), mint (15 squares). An even split would hand every color 7 squares, so the amber region is running 5 squares under that average while the mint region is the roomiest area on the grid with 15 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 2 squares while mint has 15 to spare, a gap of 13. Attack the cramped end and the roomy regions tend to resolve themselves.

Shape matters as much as size. The amber region is the most compact color here, filling 100% of the 2×1 box that encloses it, while blue is the most scattered at 44% of its 4×4 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 #176 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.

Amber occupies 2 squares within rows 1–2, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Pale yellow: 2 squares across rows 7–7, columns 6–7. It never leaves row 7, so its queen is locked to that row.

Red covers 5 squares, spanning 2 rows and 4 columns (rows 6–7, columns 4–7).

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

Blue: 7 squares across rows 2–5, columns 1–4.

Slate covers 11 squares, spanning 4 rows and 4 columns (rows 4–7, columns 1–4).

Mint occupies 15 squares within rows 1–5, columns 4–7.

Where to start on this board

Open on the tightest color. Here that is amber, confined to 2 squares between rows 1 and 2 and columns 1 and 1. Its queen has nowhere else to go, which means those 2 rows and 1 column carry more constraint than anywhere else on #176.

Because amber 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. Pale yellow, indigo 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 #176

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 6 are the narrowest, touched by only 2 colors. Columns 1, 2, 3, 5, 6 and 7 are the tightest columns, with 3 colors.

A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 6 rows. Row 2 is the crowded extreme at 4 colors — worth revisiting later, not now.

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.

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 #176 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 13. 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.

Stuck on #176 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.

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 row 3 on this board.

A non-spoiling distribution check: solved, #176 carries 3 queens on the perimeter against 4 queens inside, 1 of them cornered. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.

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 #176?

Puzzle #176 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 #176 have only one solution?

Yes. Every puzzle in the 7x7 collection, including #176, 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 #176?

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