9x9 Queens Puzzle #197
Difficulty: easy. 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 16-step reasoning is written out below.
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Step-by-step solution for puzzle #197
What follows is a complete solve of puzzle #197, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 9 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: region confined to one line, last square in a region, line confined to one region, last square in a column, proof by contradiction.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 9 squares in all.
Step 2. All of the slate region's remaining squares (2 open squares) lie in column 5. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 10 squares come off.
Step 3. Khaki cannot leave row 8 — every one of its open squares (2 open squares) sits there. That row belongs to khaki, so every square on it owned by another colour is dead: 10 squares in all.
Step 4. The orange region is down to one open square, at row 9, column 8. Every colour holds exactly one queen, so that square is forced. Row 9 and column 8 are now spoken for, along with the squares touching that queen.
The board after step 4 — 1 of 9 queens placed. Step 5. Only row 7, column 9 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 9, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 6. The khaki region is down to one open square, at row 8, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 6, the remaining khaki squares, and the ring of neighbours around row 8, column 6.
Step 7. 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: 4 squares in all.
Step 8. The slate region is down to one open square, at row 5, column 5. Every colour holds exactly one queen, so that square is forced. Row 5 and column 5 are now spoken for, along with the squares touching that queen.
Step 9. 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: 2 squares in all.
The board after step 9 — 4 of 9 queens placed. Step 10. Every open square left in column 4 (2 open squares) belongs to the amber region. Since that column must contain a queen, amber is the colour that supplies it — so amber squares anywhere else on the board are out, 3 squares in all.
Step 11. Scan column 7 and only row 4, column 7 remains — a mint square. Every column takes a queen, so it takes this one. Cross out the whole of row 4 and column 7, the remaining mint squares, and the ring of neighbours around row 4, column 7.
Step 12. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 3. Follow the forced moves that would create and column 2 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 13. 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: 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 1; amber has one square left, so its queen takes row 2, column 4; red has one square left, so its queen takes row 6, 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 6, column 3.
The board after step 13 — 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 #197
Puzzle #197 is one of the easy-rated boards in the 9x9 set. Nine colors divide its 81 squares, and you win by placing nine queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
As with every puzzle in this collection, #197 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 #197 are: indigo (2 squares), slate (2 squares), khaki (3 squares), orange (4 squares), pale yellow (10 squares), blue (11 squares), amber (15 squares), red (16 squares), mint (18 squares). An even split would hand every color 9 squares, so the indigo region is running 7 squares under that average while the mint region is the roomiest area on the grid with 18 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 indigo, with 2 squares, is under far more pressure than mint with 18. A 16-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×2 frame; pale yellow is the loosest, occupying only 40% of its 5×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Here is the full footprint of every color on puzzle #197, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Indigo occupies 2 squares within rows 1–1, columns 2–3. It never leaves row 1, so its queen is locked to that row.
Slate: 2 squares across rows 5–6, columns 5–5. It sits wholly inside column 5, so its queen must be in that column.
Khaki covers 3 squares, spanning 1 row and 3 columns (rows 8–8, columns 5–7). It never leaves row 8, so its queen is locked to that row.
Orange occupies 4 squares within rows 8–9, columns 6–8.
Pale yellow: 10 squares across rows 5–9, columns 5–9.
Blue covers 11 squares, spanning 6 rows and 3 columns (rows 1–6, columns 1–3).
Amber occupies 15 squares within rows 1–3, columns 3–8.
Red: 16 squares across rows 6–9, columns 1–5.
Mint covers 18 squares, spanning 6 rows and 7 columns (rows 1–6, columns 3–9).
Where to start on this board
The most constrained color is where a solve should begin, and on #197 that is indigo with 2 squares. It reaches from row 1 to row 1 and from column 2 to column 3 — 1 row by 2 columns. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Khaki, slate 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 #197
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 4 and 7 are the narrowest, touched by only 2 colors. Columns 1 and 9 are the tightest columns, with 2 colors.
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 8 rows. Row 6, meeting 5 colors, is the opposite case — the busiest line on #197 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.
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.
A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.
Why puzzle #197 is rated easy
The rating reflects how much the board concedes up front. Puzzle #197 has three regions confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 16 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
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 #197, start hunting for it near row 4.
Here is a shape check that gives nothing away: when #197 is solved, 4 queens sit on the outer edge of the grid and 5 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 9x9 Queens puzzle #197?
Puzzle #197 is rated easy. 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 #197 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #197, 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 #197?
Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–1 and columns 2–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.