7x7 Queens Puzzle #76
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 14-step reasoning is written out below.
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Step-by-step solution for puzzle #76
What follows is a complete solve of puzzle #76, worked the way a person works it rather than the way a computer does. It runs to 14 deductions — 7 queen placements and 7 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 proof by contradiction.
Techniques this board needs: last square in a region, region shadow, region confined to one line, line confined to one region, last square in a column, proof by contradiction.
Show the full walkthrough (11 steps) — contains the solution
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.
Step 2. The red region is down to one open square, at row 6, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 1, the remaining red squares, and the ring of neighbours around row 6, column 1.
Step 3. The indigo region has 5 squares left, spread across rows 1 and 2, columns 2, 3, 4, and 6. 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.
Step 4. Wherever blue ends up among its 2 squares, the same squares fall within reach of it — 4 squares in total, impossible regardless of how the colour resolves.
The board after step 4 — 2 of 7 queens placed. Step 5. Mint cannot leave row 4 — every one of its open squares (2 open squares) sits there. That row belongs to mint, so every square on it owned by another colour is dead: 5 squares in all.
Step 6. The amber region has 3 squares left, spread across rows 2 and 3, 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.
Step 7. Row 5 has only slate squares left (2 open squares). The row needs a queen and only slate can provide one, which pins that colour to this row and kills its squares elsewhere: 1 square.
The board after step 7 — 2 of 7 queens placed. Step 8. Column 7 now has exactly one open square, row 5, column 7, in the slate region. Columns must each hold a queen, so that placement is forced. Everything else in row 5, in column 7, in slate and in the ring around that square is now dead.
Step 9. Column 2 has only amber squares left (2 open squares). The column needs a queen and only amber can provide one, which pins that colour to this column and kills its squares elsewhere: 1 square.
Step 10. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 3. Follow the forced moves that would create and row 2 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 11. 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: column 3 is down to one square, putting a queen on row 4, column 3; amber has one square left, so its queen takes row 2, column 2; blue has one square left, so its queen takes row 3, column 6; indigo has one square left, so its queen takes row 1, column 4. 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 1, column 4.
The board after step 11 — 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 #76
This is puzzle number 76 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 #76 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 #76 are: pale yellow (1 square), mint (3 squares), blue (4 squares), red (4 squares), indigo (6 squares), amber (14 squares), slate (17 squares). An even split would hand every color 7 squares, so the pale yellow region is running 6 squares under that average while the slate region is the roomiest area on the grid with 17 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 slate with 17. A 16-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 pale yellow is the most solid, filling 100% of its 1×1 box, and indigo the most diffuse at 60% of its 2×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
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.
Mint occupies 3 squares within rows 3–4, columns 3–4.
Blue: 4 squares across rows 2–3, columns 4–6.
Red covers 4 squares, spanning 2 rows and 3 columns (rows 6–7, columns 1–3).
Indigo occupies 6 squares within rows 1–2, columns 2–6.
Amber: 14 squares across rows 1–6, columns 1–3.
Slate covers 17 squares, spanning 7 rows and 4 columns (rows 1–7, 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 #76.
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. Indigo, 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 #76
A quick way to find the next move is to count how many different colors touch each line. On this board, row 5 is the narrowest, touched by only 2 colors. Column 7 is the tightest column, with 1 color.
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 #76 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.
Resist the plausible-looking queen. On #76 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 #76 is rated medium
Ratings track how generous the opening is. On #76 you get two regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 16-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.
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.
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 #76, start hunting for it near column 7.
Here is a shape check that gives nothing away: when #76 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
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- 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.
- 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 #76?
Puzzle #76 is rated medium. Its smallest color region holds 1 square, 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 #76 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #76, 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 #76?
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.