9x9 Queens Puzzle #205
Difficulty: medium. 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 29-step reasoning is written out below.
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Step-by-step solution for puzzle #205
What follows is a complete solve of puzzle #205, worked the way a person works it rather than the way a computer does. It runs to 29 deductions — 9 queen placements and 20 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, pairing argument on rows, pairing argument on columns, proof by contradiction.
Show the full walkthrough (24 steps) — contains the solution
Step 1. Only row 2, column 1 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 1, the rest of the amber region and every square adjacent to it — diagonals included.
Step 2. Mint is down to 6 candidates in rows 1, 3, 4, 5, and 6, columns 8 and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.
Step 3. 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: 10 squares in all.
Step 4. Wherever khaki ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Column 2 has only slate squares left (5 open squares). The column needs a queen and only slate can provide one, which pins that colour to this column and kills its squares elsewhere: 6 squares.
Step 6. Here is the pairing argument. The pale yellow, khaki, and orange regions have no open squares outside rows 7–9 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 3 squares come off in one stroke.
The board after step 6 — 1 of 9 queens placed. Step 7. Mint and khaki are now boxed into columns 8 and 9 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 3 squares at once.
Step 8. Pale yellow is down to 4 candidates in rows 7–9, columns 5–7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 9. Test row 1, column 4 for indigo: place a queen there, run the forced consequences through, and column 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 10. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 5. Follow the forced moves that would create and the pale yellow region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 11. Test row 1, column 6 for indigo: place a queen there, run the forced consequences through, and column 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 12. Now a short proof by contradiction. Suppose the indigo queen went to row 3, column 4. Follow the forced moves that would create and row 8 runs out of open squares entirely — impossible. So that square is ruled out for good.
The board after step 12 — 1 of 9 queens placed. Step 13. Test row 3, column 5 for indigo: place a queen there, run the forced consequences through, and the pale yellow region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 14. The indigo region is down to one open square, at row 1, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 3, the remaining indigo squares, and the ring of neighbours around row 1, column 3.
Step 15. Test row 3, column 7 for blue: place a queen there, run the forced consequences through, and the pale yellow region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 16. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 5. Follow the forced moves that would create and the pale yellow region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 17. Test row 4, column 6 for blue: place a queen there, run the forced consequences through, and row 8 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 18. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 7. Follow the forced moves that would create and the pale yellow region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
The board after step 18 — 2 of 9 queens placed. Step 19. Test row 5, column 4 for blue: place a queen there, run the forced consequences through, and the orange region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 20. Blue is down to 2 candidates in rows 3 and 5, columns 5 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 21. Only row 6, column 7 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 7, the rest of the red region and every square adjacent to it — diagonals included.
Step 22. Wherever pale yellow ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 23. Test row 5, column 5 for blue: place a queen there, run the forced consequences through, and the khaki region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 24. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: blue has one square left, so its queen takes row 3, column 6; pale yellow has one square left, so its queen takes row 7, column 5; orange has one square left, so its queen takes row 9, column 4; khaki has one square left, so its queen takes row 8, column 9; mint has one square left, so its queen takes row 4, column 8; slate has one square left, so its queen takes row 5, column 2. 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 5, column 2.
The board after step 24 — 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 #205
Puzzle #205 is one of the medium-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.
There is exactly one way to finish #205, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these nine regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #205 are: amber (1 square), red (2 squares), khaki (6 squares), mint (8 squares), pale yellow (8 squares), orange (9 squares), indigo (11 squares), blue (16 squares), slate (20 squares). An even split would hand every color 9 squares, so the amber region is running 8 squares under that average while the slate region is the roomiest area on the grid with 20 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 amber, with 1 square, is under far more pressure than slate with 20. A 19-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Amber is the densest region on this grid at 100% of a 1×1 frame; blue is the loosest, occupying only 53% of its 6×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Below is every color on #205 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.
Amber covers 1 square, spanning 1 row and 1 column (rows 2–2, columns 1–1). It never leaves row 2, so its queen is locked to that row.
Red occupies 2 squares within rows 6–6, columns 6–7. It never leaves row 6, so its queen is locked to that row.
Khaki: 6 squares across rows 5–9, columns 8–9.
Mint covers 8 squares, spanning 6 rows and 2 columns (rows 1–6, columns 8–9).
Pale yellow occupies 8 squares within rows 7–9, columns 5–8.
Orange: 9 squares across rows 6–9, columns 3–6.
Indigo covers 11 squares, spanning 3 rows and 6 columns (rows 1–3, columns 1–6).
Blue occupies 16 squares within rows 1–6, columns 4–8.
Slate: 20 squares across rows 3–9, columns 1–4.
Where to start on this board
The most constrained color is where a solve should begin, and on #205 that is amber with 1 square. It reaches from row 2 to row 2 and from column 1 to column 1 — 1 row by 1 column. 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 amber never leaves row 2, its queen must be in that row — and that instantly settles row 2 for everyone else. Sweep along it and X out every square that is not amber. 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. Red is no more than two rows tall, so its queen is 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 #205
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 4 are the narrowest, touched by only 3 colors. Columns 2 and 9 are the tightest columns, 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 8 rows. Row 6 is the crowded extreme at 6 colors — worth revisiting later, not now.
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 #205 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers three single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 19. 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.
Stuck on #205 despite the medium rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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 #205, start hunting for it near column 2.
A non-spoiling distribution check: solved, #205 carries 4 queens on the perimeter against 5 queens inside, with the corners empty. 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 9x9 Queens puzzle #205?
Puzzle #205 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #205 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #205, 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 #205?
Start with the amber region — the most constrained color on this board, with 1 square spanning rows 2–2 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.