7x7 Queens Puzzle #80
Difficulty: hard. 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 13-step reasoning is written out below.
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Step-by-step solution for puzzle #80
What follows is a complete solve of puzzle #80, worked the way a person works it rather than the way a computer does. It runs to 13 deductions — 7 queen placements and 6 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: region confined to one line, last square in a region, last square in a column.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Indigo cannot leave column 1 — every one of its open squares (3 open squares) sits there. That column belongs to indigo, so every square in it owned by another colour is dead: 5 squares in all.
Step 2. The mint region is down to one open square, at row 4, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 2, the remaining mint squares, and the ring of neighbours around row 4, column 2.
Step 3. Amber cannot leave row 2 — every one of its open squares (2 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 7 squares in all.
Step 4. The indigo region is down to one open square, at row 1, column 1. Every colour holds exactly one queen, so that square is forced. Row 1 and column 1 are now spoken for, along with the squares touching that queen.
The board after step 4 — 2 of 7 queens placed. Step 5. Blue cannot leave row 3 — every one of its open squares (2 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 1 square in all.
Step 6. All of the slate region's remaining squares (3 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 7. Only row 3, column 5 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 5, in blue and in the ring around that square is now dead.
Step 8. The amber region is down to one open square, at row 2, column 3. Every colour holds exactly one queen, so that square is forced. Row 2 and column 3 are now spoken for, along with the squares touching that queen.
Step 9. 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: 1 square in all.
The board after step 9 — 4 of 7 queens placed. Step 10. All of the slate region's remaining squares (2 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
Step 11. Scan column 6 and only row 7, column 6 remains — a pale yellow square. Every column takes a queen, so it takes this one. Cross out the whole of row 7 and column 6, the remaining pale yellow squares, and the ring of neighbours around row 7, column 6.
Step 12. The slate region is down to one open square, at row 5, column 7. Every colour holds exactly one queen, so that square is forced. Row 5 and column 7 are now spoken for, along with the squares touching that queen.
Step 13. Only row 6, column 4 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 4, the rest of the red region and every square adjacent to it — diagonals included.
The board after step 13 — 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 #80
This is puzzle number 80 in the 7x7 collection, rated hard. 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.
As with every puzzle in this collection, #80 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 #80 are: indigo (3 squares), mint (3 squares), pale yellow (4 squares), slate (6 squares), amber (7 squares), blue (11 squares), red (15 squares). An even split would hand every color 7 squares, so the indigo region is running 4 squares under that average while the red region is the roomiest area on the grid with 15 squares.
Every color takes one queen and one only, whatever its size — which is why the 12 square difference between indigo 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 indigo region is the most compact color here, filling 100% of the 3×1 box that encloses it, while red is the most scattered at 50% of its 5×6 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 #80 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.
Indigo occupies 3 squares within rows 1–3, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Mint: 3 squares across rows 4–5, columns 1–2.
Pale yellow covers 4 squares, spanning 2 rows and 3 columns (rows 6–7, columns 4–6).
Slate occupies 6 squares within rows 3–7, columns 6–7.
Amber: 7 squares across rows 1–4, columns 2–4.
Blue covers 11 squares, spanning 4 rows and 5 columns (rows 1–4, columns 3–7).
Red occupies 15 squares within rows 3–7, columns 1–6.
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 3 squares between rows 1 and 3 and columns 1 and 1. Its queen has nowhere else to go, which means those 3 rows and 1 column carry more constraint than anywhere else on #80.
Because indigo 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. Mint 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 #80
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 5, 6 and 7 are the narrowest, touched by only 3 colors. Column 7 is the tightest column, with 2 colors.
A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 6 rows. Row 4 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
The middle of #80 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.
The whole middle game reduces to two sweeps. Sweep the seven colors looking for one with a single square left; sweep the seven rows and seven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 7x7 solve from a stalled one.
Resist the plausible-looking queen. On #80 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 #80 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers one single-line region, a 3-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 12. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
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.
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 #80 is column 7.
A non-spoiling distribution check: solved, #80 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
- 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 #80?
Puzzle #80 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 7x7 puzzle #80 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #80, 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 #80?
Start with the indigo region — the most constrained color on this board, with 3 squares spanning rows 1–3 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.