7x7 Queens Puzzle #4

Difficulty: easy. 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 9-step reasoning is written out below.

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Step-by-step solution for puzzle #4

What follows is a complete solve of puzzle #4, worked the way a person works it rather than the way a computer does. It runs to 9 deductions — 7 queen placements and 2 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 confined to one line.

Techniques this board needs: last square in a region, region confined to one line.

Show the full walkthrough (5 steps) — contains the solution

  1. Step 1. Only row 7, column 5 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 5, the rest of the pale yellow region and every square adjacent to it — diagonals included.

  2. Step 2. The red region is down to one open square, at row 6, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 7, the remaining red squares, and the ring of neighbours around row 6, column 7.

  3. Step 3. Blue cannot leave column 3 — 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.

    Puzzle #4 after step 3: 2 of 7 queens placed
    The board after step 3 — 2 of 7 queens placed.

  4. Step 4. All of the mint region's remaining squares (2 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.

  5. Step 5. 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: 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 3; slate has one square left, so its queen takes row 5, column 4; mint has one square left, so its queen takes row 4, 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 #4 after step 5: 7 of 7 queens placed
    The board after step 5 — 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 #4

This is puzzle number 4 in the 7x7 collection, rated easy. 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 #4, 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 #4 are: pale yellow (1 square), blue (2 squares), mint (2 squares), red (5 squares), slate (10 squares), indigo (13 squares), amber (16 squares). An even split would hand every color 7 squares, so the pale yellow region is running 6 squares under that average while the amber region is the roomiest area on the grid with 16 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 amber with 16. A 15-square gap like this one rewards players who start where the choices are fewest.

A region's outline can constrain it more than its size does. Here blue is the most solid, filling 100% of its 2×1 box, and indigo the most diffuse at 62% of its 7×3 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

Region by region

Here is the full footprint of every color on puzzle #4, 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 5–5). It never leaves row 7, so its queen is locked to that row.

Blue occupies 2 squares within rows 2–3, columns 3–3. It sits wholly inside column 3, so its queen must be in that column.

Mint: 2 squares across rows 4–5, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

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

Slate occupies 10 squares within rows 4–7, columns 3–5.

Indigo: 13 squares across rows 1–7, columns 1–3.

Amber covers 16 squares, spanning 5 rows and 4 columns (rows 1–5, columns 4–7).

Where to start on this board

Open on the tightest color. Here that is pale yellow, confined to 1 square between rows 7 and 7 and columns 5 and 5. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #4.

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.

Work outward to the other flat regions next. Blue, mint 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 #4

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. Column 2 is the tightest column, with 1 color.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 6 rows. By contrast row 7 meets 4 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

Working through the middle game

The middle of #4 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.

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.

Resist the plausible-looking queen. On #4 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.

Why puzzle #4 is rated easy

The rating reflects how much the board concedes up front. Puzzle #4 has four 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 15 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 #4 despite the easy 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 #4, start hunting for it near column 2.

As a sanity check rather than a spoiler: the finished solution to #4 puts 4 queens on the outer border and 3 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.

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 #4?

Puzzle #4 is rated easy. 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 #4 have only one solution?

Yes. Every puzzle in the 7x7 collection, including #4, 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 #4?

Start with the pale yellow region — the most constrained color on this board, with 1 square spanning rows 7–7 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.