9x9 Queens Puzzle #240
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 14-step reasoning is written out below.
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Step-by-step solution for puzzle #240
What follows is a complete solve of puzzle #240, worked the way a person works it rather than the way a computer does. It runs to 14 deductions — 9 queen placements and 5 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 shadow, pairing argument on rows, pairing argument on columns, region confined to one line.
Show the full walkthrough (10 steps) — contains the solution
Step 1. Only row 4, column 6 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 6, the rest of the mint region and every square adjacent to it — diagonals included.
Step 2. The indigo region is down to one open square, at row 1, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 5, the remaining indigo squares, and the ring of neighbours around row 1, column 5.
Step 3. The blue region has 3 squares left, spread across rows 2 and 3, columns 1 and 2. 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 3 — 2 of 9 queens placed. Step 4. Wherever red ends up among its 3 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
Step 5. Amber and blue are now boxed into rows 2 and 3 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 3 squares at once.
Step 6. Here is the pairing argument. The blue and orange regions have no open squares outside columns 1 and 2 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 3 squares come off in one stroke.
Step 7. Only row 6, column 3 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 3, in red and in the ring around that square is now dead.
The board after step 7 — 3 of 9 queens placed. Step 8. The slate region is down to one open square, at row 5, column 8. Every colour holds exactly one queen, so that square is forced. Row 5 and column 8 are now spoken for, along with the squares touching that queen.
Step 9. Pale yellow cannot leave column 9 — every one of its open squares (3 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 2 squares in all.
Step 10. 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 7; blue has one square left, so its queen takes row 3, column 1; orange has one square left, so its queen takes row 9, column 2; khaki has one square left, so its queen takes row 8, column 4; pale yellow has one square left, so its queen takes row 7, column 9. 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 7, column 9.
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 #240
This is puzzle number 240 in the 9x9 collection, rated medium. The grid holds 81 squares split between nine colored regions, and a finished board carries nine queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
There is exactly one way to finish #240, 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 #240 are: mint (1 square), indigo (2 squares), red (3 squares), orange (6 squares), blue (7 squares), pale yellow (8 squares), amber (15 squares), khaki (17 squares), slate (22 squares). An even split would hand every color 9 squares, so the mint region is running 8 squares under that average while the slate region is the roomiest area on the grid with 22 squares.
Every color takes one queen and one only, whatever its size — which is why the 21 square difference between mint and slate is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #240, packing 100% of its 1×2 bounding box, whereas slate straggles across its 5×8 box at just 55%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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 medium board.
Mint: 1 square across rows 4–4, columns 6–6. It never leaves row 4, so its queen is locked to that row.
Indigo covers 2 squares, spanning 1 row and 2 columns (rows 1–1, columns 5–6). It never leaves row 1, so its queen is locked to that row.
Red occupies 3 squares within rows 6–7, columns 2–3.
Orange: 6 squares across rows 5–9, columns 1–2.
Blue covers 7 squares, spanning 3 rows and 4 columns (rows 1–3, columns 1–4).
Pale yellow occupies 8 squares within rows 5–9, columns 8–9.
Amber: 15 squares across rows 1–4, columns 5–9.
Khaki covers 17 squares, spanning 4 rows and 6 columns (rows 6–9, columns 2–7).
Slate occupies 22 squares within rows 2–6, columns 1–8.
Where to start on this board
Begin where the choices are fewest. On this board that means mint, which owns just 1 square inside a 1×1 window running from row 4 to row 4 and column 6 to column 6. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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 red 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 #240
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, 4, 5, 8 and 9 are the narrowest, touched by only 3 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 3 colors, so one of that handful must take the row and the rest are squeezed into the other 8 rows. Row 6 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
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.
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 #240 is rated medium
The medium label comes out of four measurable things about #240: three 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 21. 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.
Stuck on #240 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.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #240 is column 9.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 5 queens in the middle, and nothing in the corners. 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 #240?
Puzzle #240 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 #240 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #240, 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 #240?
Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.