9x9 Queens Puzzle #75
Difficulty: hard. Place 9 queens on this 9x9 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 #75
What follows is a complete solve of puzzle #75, worked the way a person works it rather than the way a computer does. It runs to 12 deductions — 9 queen placements and 3 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, last square in a region, region shadow.
Show the full walkthrough (8 steps) — contains the solution
Step 1. Amber cannot leave column 9 — 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: 9 squares in all.
Step 2. The mint region is down to one open square, at row 4, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 8, the remaining mint squares, and the ring of neighbours around row 4, column 8.
Step 3. Only row 2, column 9 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 9, in amber and in the ring around that square is now dead.
Step 4. All of the indigo region's remaining squares (3 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.
The board after step 4 — 2 of 9 queens placed. Step 5. Only row 3, column 5 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 5, the rest of the blue region and every square adjacent to it — diagonals included.
Step 6. The slate region is down to one open square, at row 5, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 6, the remaining slate squares, and the ring of neighbours around row 5, column 6.
Step 7. Wherever red ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 8. 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: khaki has one square left, so its queen takes row 8, column 1; red has one square left, so its queen takes row 6, column 2; indigo has one square left, so its queen takes row 1, column 3; orange has one square left, so its queen takes row 9, column 4; pale yellow has one square left, so its queen takes row 7, column 7. 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 7, column 7.
The board after step 8 — 9 of 9 queens placed.
That completes the board: 9 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 9x9 Queens puzzle #75
9x9 puzzle #75 is rated hard. Nine colored regions cover its 81 squares, and your job is to seat nine queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Puzzle #75 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 #75 are: amber (2 squares), mint (2 squares), khaki (2 squares), indigo (6 squares), slate (10 squares), orange (10 squares), blue (13 squares), red (13 squares), pale yellow (23 squares). An even split would hand every color 9 squares, so the amber region is running 7 squares under that average while the pale yellow region is the roomiest area on the grid with 23 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Amber has only 2 places to put its queen; pale yellow has 23. The spread between them on this board is 21 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
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 red is the most scattered at 43% of its 6×5 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
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.
Amber: 2 squares across rows 2–3, columns 9–9. It sits wholly inside column 9, so its queen must be in that column.
Mint covers 2 squares, spanning 1 row and 2 columns (rows 4–4, columns 8–9). It never leaves row 4, so its queen is locked to that row.
Khaki occupies 2 squares within rows 7–8, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Indigo: 6 squares across rows 1–2, columns 1–4.
Slate covers 10 squares, spanning 3 rows and 5 columns (rows 3–5, columns 3–7).
Orange occupies 10 squares within rows 8–9, columns 1–6.
Blue: 13 squares across rows 1–3, columns 4–9.
Red covers 13 squares, spanning 6 rows and 5 columns (rows 2–7, columns 1–5).
Pale yellow occupies 23 squares within rows 5–9, columns 3–9.
Where to start on this board
Amber is the natural opening on #75: it is the smallest region on the grid at 2 squares, boxed into rows 2–3 and columns 9–9. 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 amber stays inside column 9, its queen belongs to that column and no other region may use it. Cross out the rest of column 9 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, indigo, khaki and orange 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 #75
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 6 and 9 are the narrowest, touched by only 2 colors. Columns 2, 7 and 8 are the tightest columns, with 3 colors.
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 8 rows. By contrast row 3 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 #75 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 81 squares of this grid. First: has any of the nine regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 9x9 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 #75 is rated hard
Ratings track how generous the opening is. On #75 you get three regions pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 2 colors, and a 21-square gap between the largest and smallest regions. Those four numbers largely explain the hard 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 9x9
At 9x9 a full re-scan after every move starts to get slow, so work locally instead. Once a queen goes down, check only the regions, rows and columns that placement actually touched — that is where new information appeared. Fall back to a complete sweep only when the local pass turns up nothing, and your solve times will drop noticeably.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #75 the place to look for one is around row 1.
As a sanity check rather than a spoiler: the finished solution to #75 puts 4 queens on the outer border and 5 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
- 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 9x9 Queens puzzle #75?
Puzzle #75 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #75 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #75, has been verified to have exactly one valid arrangement of nine queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #75?
Start with the amber region — the most constrained color on this board, with 2 squares spanning rows 2–3 and columns 9–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.