9x9 Queens Puzzle #2
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 13-step reasoning is written out below.
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Step-by-step solution for puzzle #2
What follows is a complete solve of puzzle #2, worked the way a person works it rather than the way a computer does. It runs to 13 deductions — 9 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: last square in a region, region shadow, region confined to one line.
Show the full walkthrough (10 steps) — contains the solution
Step 1. Only row 5, column 2 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 2, the rest of the slate region and every square adjacent to it — diagonals included.
Step 2. 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. Cross out the whole of row 1 and column 1, the remaining indigo squares, and the ring of neighbours around row 1, column 1.
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.
The board after step 3 — 3 of 9 queens placed. Step 4. Wherever blue ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Red cannot leave row 6 — every one of its open squares (2 open squares) sits there. That row belongs to red, so every square on it owned by another colour is dead: 5 squares in all.
Step 6. All of the pale yellow 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: 6 squares come off.
Step 7. Only row 3, column 6 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 6, in blue and in the ring around that square is now dead.
The board after step 7 — 4 of 9 queens placed. Step 8. 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.
Step 9. Khaki cannot leave column 3 — every one of its open squares (2 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 3 squares in all.
Step 10. 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: mint has one square left, so its queen takes row 4, column 4; orange has one square left, so its queen takes row 9, column 8; pale yellow has one square left, so its queen takes row 7, column 7; khaki has one square left, so its queen takes row 8, column 3. 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 8, column 3.
The board after step 10 — 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 #2
Board #2 of the 9x9 bank carries a hard rating. Its 81 squares are shared between nine colored regions, and the solution places nine queens: one per row, one per column, one per region, none of them side by side or corner to corner.
As with every puzzle in this collection, #2 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 #2 are: slate (1 square), indigo (3 squares), amber (3 squares), pale yellow (4 squares), red (5 squares), blue (12 squares), orange (13 squares), khaki (16 squares), mint (24 squares). An even split would hand every color 9 squares, so the slate region is running 8 squares under that average while the mint region is the roomiest area on the grid with 24 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Slate is working with 1 square while mint has 24 to spare, a gap of 23. Attack the cramped end and the roomy regions tend to resolve themselves.
Shape matters as much as size. The slate region is the most compact color here, filling 100% of the 1×1 box that encloses it, while mint is the most scattered at 53% of its 9×5 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
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a hard board.
Slate occupies 1 square within rows 5–5, columns 2–2. It never leaves row 5, so its queen is locked to that row.
Indigo: 3 squares across rows 1–2, columns 1–2.
Amber covers 3 squares, spanning 2 rows and 2 columns (rows 1–2, columns 8–9).
Pale yellow occupies 4 squares within rows 6–8, columns 6–7.
Red: 5 squares across rows 4–6, columns 5–6.
Blue covers 12 squares, spanning 5 rows and 4 columns (rows 1–5, columns 6–9).
Orange occupies 13 squares within rows 4–9, columns 6–9.
Khaki: 16 squares across rows 2–9, columns 1–3.
Mint covers 24 squares, spanning 9 rows and 5 columns (rows 1–9, columns 2–6).
Where to start on this board
Open on the tightest color. Here that is slate, confined to 1 square between rows 5 and 5 and columns 2 and 2. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #2.
Because slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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 and amber 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 #2
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 9 are the narrowest, touched by only 3 colors. Column 4 is the tightest column, with 1 color.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 8 rows. Row 5 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.
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.
The whole middle game reduces to two sweeps. Sweep the nine colors looking for one with a single square left; sweep the nine rows and nine columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 9x9 solve from a stalled one.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #2 is rated hard
The hard label comes out of four measurable things about #2: two of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 23. Boards that score tight on those measures give up their first queen fast.
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.
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 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.
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 column 4 on this board.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 6 queens in the middle, including 1 in a corner. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #2?
Puzzle #2 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #2 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #2, 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 #2?
Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.