9x9 Queens Puzzle #138
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 #138
What follows is a complete solve of puzzle #138, 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 confined to one line, region shadow, last square in a column.
Show the full walkthrough (9 steps) — contains the solution
Step 1. Only row 9, column 9 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 9, the rest of the orange region and every square adjacent to it — diagonals included.
Step 2. All of the slate region's remaining squares (2 open squares) lie in column 6. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 10 squares come off.
Step 3. Only row 6, column 8 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 8, in red and in the ring around that square is now dead.
The board after step 3 — 2 of 9 queens placed. Step 4. 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. Row 5 and column 6 are now spoken for, along with the squares touching that queen.
Step 5. Only row 3, column 7 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 7, the rest of the blue region and every square adjacent to it — diagonals included.
Step 6. The pale yellow region has 4 squares left, spread across rows 7 and 8, columns 3–5. 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.
The board after step 6 — 4 of 9 queens placed. Step 7. Wherever khaki 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. All of the mint region's remaining squares (2 open squares) lie in row 4. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 9. 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: amber has one square left, so its queen takes row 2, column 5; column 4 is down to one square, putting a queen on row 7, column 4; khaki has one square left, so its queen takes row 8, column 1; mint has one square left, so its queen takes row 4, column 2; indigo has one square left, so its queen takes row 1, 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 1, column 3.
The board after step 9 — 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 #138
Board #138 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.
Like every board on this site, #138 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 #138 are: orange (1 square), slate (2 squares), khaki (8 squares), indigo (9 squares), red (9 squares), blue (11 squares), mint (12 squares), amber (14 squares), pale yellow (15 squares). An even split would hand every color 9 squares, so the orange region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 15 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 pale yellow with 15. A 14-square gap like this one rewards players who start where the choices are fewest.
Shape matters as much as size. The slate region is the most compact color here, filling 100% of the 2×1 box that encloses it, while amber is the most scattered at 47% of its 5×6 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.
Orange: 1 square across rows 9–9, columns 9–9. It never leaves row 9, so its queen is locked to that row.
Slate covers 2 squares, spanning 2 rows and 1 column (rows 5–6, columns 6–6). It sits wholly inside column 6, so its queen must be in that column.
Khaki occupies 8 squares within rows 6–9, columns 1–4.
Indigo: 9 squares across rows 1–2, columns 1–5.
Red covers 9 squares, spanning 5 rows and 3 columns (rows 5–9, columns 7–9).
Blue occupies 11 squares within rows 1–5, columns 7–9.
Mint: 12 squares across rows 3–8, columns 1–3.
Amber covers 14 squares, spanning 5 rows and 6 columns (rows 1–5, columns 3–8).
Pale yellow occupies 15 squares within rows 5–9, columns 3–8.
Where to start on this board
Open on the tightest color. Here that is orange, confined to 1 square between rows 9 and 9 and columns 9 and 9. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #138.
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. Indigo and slate 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 #138
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3 and 4 are the narrowest, touched by only 3 colors. Columns 1, 2, 5, 6 and 9 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 8 rows. Row 5, meeting 6 colors, is the opposite case — the busiest line on #138 and the last one you should expect to crack.
Working through the middle game
The middle of #138 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.
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 #138 is rated hard
Ratings track how generous the opening is. On #138 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 14-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.
If #138 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 #138, start hunting for it near row 1.
Here is a shape check that gives nothing away: when #138 is solved, 3 queens sit on the outer edge of the grid and 6 sit inside it, with 1 queen 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 #138?
Puzzle #138 is rated hard. 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 #138 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #138, 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 #138?
Start with the orange region — the most constrained color on this board, with 1 square spanning rows 9–9 and columns 9–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.