9x9 Queens Puzzle #41
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 17-step reasoning is written out below.
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Step-by-step solution for puzzle #41
What follows is a complete solve of puzzle #41, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 9 queen placements and 8 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: region shadow, region confined to one line, line confined to one region, pairing argument on columns, last square in a row, last square in a column, last square in a region.
Show the full walkthrough (9 steps) — contains the solution
Step 1. Wherever blue ends up among its 6 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 2. Pale yellow is down to 4 candidates in rows 7 and 8, columns 3–5, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 3. Orange cannot leave row 9 — every one of its open squares (2 open squares) sits there. That row belongs to orange, so every square on it owned by another colour is dead: 9 squares in all.
The board after step 3 — 0 of 9 queens placed. Step 4. All of the khaki region's remaining squares (2 open squares) lie in row 8. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.
Step 5. Pale yellow cannot leave row 7 — every one of its open squares (2 open squares) sits there. That row belongs to pale yellow, so every square on it owned by another colour is dead: 6 squares in all.
Step 6. The blue region has 4 squares left, spread across rows 3–5, 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.
The board after step 6 — 0 of 9 queens placed. Step 7. Row 1 has only indigo squares left (9 open squares). The row needs a queen and only indigo can provide one, which pins that colour to this row and kills its squares elsewhere: 9 squares.
Step 8. Here is the pairing argument. The red 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: 6 squares come off in one stroke.
Step 9. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: row 2 is down to one square, putting a queen on row 2, column 9; column 8 is down to one square, putting a queen on row 4, column 8; slate has one square left, so its queen takes row 5, column 3; blue has one square left, so its queen takes row 3, column 5; red has one square left, so its queen takes row 6, column 1; pale yellow has one square left, so its queen takes row 7, column 4; khaki has one square left, so its queen takes row 8, column 6; indigo has one square left, so its queen takes row 1, column 7; orange has one square left, so its queen takes row 9, 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 9, column 2.
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 #41
Puzzle #41 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.
As with every puzzle in this collection, #41 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 #41 are: orange (2 squares), pale yellow (5 squares), blue (6 squares), amber (7 squares), slate (7 squares), khaki (7 squares), red (9 squares), indigo (18 squares), mint (20 squares). An even split would hand every color 9 squares, so the orange region is running 7 squares under that average while the mint 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 orange, with 2 squares, is under far more pressure than mint with 20. A 18-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Orange is the densest region on this grid at 100% of a 1×2 frame; blue is the loosest, occupying only 50% of its 4×3 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 #41, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Orange occupies 2 squares within rows 9–9, columns 1–2. It never leaves row 9, so its queen is locked to that row.
Pale yellow: 5 squares across rows 6–8, columns 3–5.
Blue covers 6 squares, spanning 4 rows and 3 columns (rows 3–6, columns 3–5).
Amber occupies 7 squares within rows 2–6, columns 8–9.
Slate: 7 squares across rows 2–5, columns 2–4.
Khaki covers 7 squares, spanning 2 rows and 4 columns (rows 8–9, columns 3–6).
Red occupies 9 squares within rows 3–8, columns 1–2.
Indigo: 18 squares across rows 1–3, columns 1–9.
Mint covers 20 squares, spanning 7 rows and 4 columns (rows 3–9, columns 6–9).
Where to start on this board
The most constrained color is where a solve should begin, and on #41 that is orange with 2 squares. It reaches from row 9 to row 9 and from column 1 to column 2 — 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 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. Khaki 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 #41
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 1 color. Column 7 is the tightest column, with 2 colors.
A line with few colors is a line close to resolving. Where only 1 color appear, one of them owns the row outright and the others must fit into the remaining 8 rows. Row 3 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.
Resist the plausible-looking queen. On #41 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 #41 is rated medium
The rating reflects how much the board concedes up front. Puzzle #41 has one region confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 1 color, and the gap between its largest and smallest regions is 18 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
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 #41, start hunting for it near row 1.
A non-spoiling distribution check: solved, #41 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 #41?
Puzzle #41 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 1 color, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #41 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #41, 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 #41?
Start with the orange region — the most constrained color on this board, with 2 squares spanning rows 9–9 and columns 1–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.