9x9 Queens Puzzle #145
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 18-step reasoning is written out below.
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Step-by-step solution for puzzle #145
What follows is a complete solve of puzzle #145, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 9 queen placements and 9 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 pairing argument on columns.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on columns, pairing argument on rows, last square in a column.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Only row 9, column 3 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 3, the rest of the orange region and every square adjacent to it — diagonals included.
Step 2. All of the indigo region's remaining squares (3 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 6 squares come off.
Step 3. Only row 3, column 8 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 8, in blue and in the ring around that square is now dead.
Step 4. The indigo region is down to one open square, at row 1, column 9. Every colour holds exactly one queen, so that square is forced. Row 1 and column 9 are now spoken for, along with the squares touching that queen.
Step 5. Amber cannot leave row 2 — every one of its open squares (3 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 2 squares in all.
The board after step 5 — 3 of 9 queens placed. Step 6. The pale yellow region has 2 squares left, spread across rows 7 and 8, columns 1 and 2. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 7. Wherever slate 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. Mint is down to 5 candidates in rows 4–6, columns 4, 5, and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.
Step 9. Slate and pale yellow are now boxed into columns 1 and 2 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 2 squares at once.
The board after step 9 — 3 of 9 queens placed. Step 10. The amber region is down to one open square, at row 2, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 6, the remaining amber squares, and the ring of neighbours around row 2, column 6.
Step 11. Red cannot leave column 7 — every one of its open squares (2 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 2 squares in all.
Step 12. The khaki region has 4 squares left, spread across rows 6–8, columns 4 and 5. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.
Step 13. Red, pale yellow, and khaki are now boxed into rows 6–8 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 1 square at once.
Step 14. 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: column 2 is down to one square, putting a queen on row 7, column 2; red has one square left, so its queen takes row 6, column 7; khaki has one square left, so its queen takes row 8, column 5; mint has one square left, so its queen takes row 4, column 4; slate has one square left, so its queen takes row 5, column 1. 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 1.
The board after step 14 — 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 #145
Puzzle #145 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.
Like every board on this site, #145 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.
How the colors divide this board
From smallest to largest, the regions on puzzle #145 are: orange (1 square), blue (2 squares), indigo (3 squares), pale yellow (7 squares), slate (8 squares), khaki (11 squares), amber (13 squares), mint (17 squares), red (19 squares). An even split would hand every color 9 squares, so the orange region is running 8 squares under that average while the red region is the roomiest area on the grid with 19 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 orange, with 1 square, is under far more pressure than red with 19. A 18-square gap like this one rewards players who start where the choices are fewest.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 3×1 box, and slate the most diffuse at 53% of its 5×3 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #145 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.
Orange covers 1 square, spanning 1 row and 1 column (rows 9–9, columns 3–3). It never leaves row 9, so its queen is locked to that row.
Blue occupies 2 squares within rows 2–3, columns 8–8. It sits wholly inside column 8, so its queen must be in that column.
Indigo: 3 squares across rows 1–3, columns 9–9. It sits wholly inside column 9, so its queen must be in that column.
Pale yellow covers 7 squares, spanning 3 rows and 3 columns (rows 7–9, columns 1–3).
Slate occupies 8 squares within rows 3–7, columns 1–3.
Khaki: 11 squares across rows 5–9, columns 4–7.
Amber covers 13 squares, spanning 2 rows and 8 columns (rows 1–2, columns 1–8).
Mint occupies 17 squares within rows 2–6, columns 2–7.
Red: 19 squares across rows 4–9, columns 6–9.
Where to start on this board
The most constrained color is where a solve should begin, and on #145 that is orange with 1 square. It reaches from row 9 to row 9 and from column 3 to column 3 — 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 orange never leaves row 9, its queen must be in that row — and that instantly settles row 9 for everyone else. Sweep along it and X out every square that is not orange. 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. Amber and blue 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 #145
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Column 9 is the tightest column, with 2 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 colors, so one of that handful must take the row and the rest are squeezed into the other 8 rows. Row 9 sits at the far end with 4 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
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.
Resist the plausible-looking queen. On #145 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 #145 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers four single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 18. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
If #145 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.
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 #145, start hunting for it near row 1.
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 #145?
Puzzle #145 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #145 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #145, 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 #145?
Start with the orange region — the most constrained color on this board, with 1 square spanning rows 9–9 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.