11x11 Queens Puzzle #6
Difficulty: hard. Place 11 queens on this 11x11 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 24-step reasoning is written out below.
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Step-by-step solution for puzzle #6
What follows is a complete solve of puzzle #6, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 11 queen placements and 13 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, pairing argument on rows, pairing argument on columns, last square in a region, last square in a row.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Wherever indigo 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 2. Mint is down to 5 candidates in rows 2–4, columns 9–11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.
Step 3. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 8 squares in all.
Step 4. Wherever amber ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
The board after step 4 — 0 of 11 queens placed. Step 5. Pale yellow is down to 5 candidates in rows 5–7, columns 2 and 3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 6. The teal region has 3 squares left, spread across rows 10 and 11, columns 4 and 5. 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. Amber, blue, and mint are now boxed into rows 2–4 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 8 squares at once.
Step 8. All of the red region's remaining squares (2 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 7 squares come off.
Step 9. The pale yellow region has 4 squares left, spread across rows 5–7, columns 2 and 3. 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 9 — 0 of 11 queens placed. Step 10. Here is the pairing argument. The slate, red, and pale yellow regions have no open squares outside rows 5–7 — 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: 10 squares come off in one stroke.
Step 11. Amber and pale yellow are now boxed into columns 2 and 3 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 7 squares at once.
Step 12. All of the orange region's remaining squares (2 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 8 squares come off.
Step 13. Only row 11, column 5 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 5, the rest of the teal region and every square adjacent to it — diagonals included.
The board after step 13 — 1 of 11 queens placed. Step 14. Row 4 has a single open square left, at column 10 in the mint region. Every row needs a queen, so this one is forced. Row 4 and column 10 are now spoken for, along with the squares touching that queen.
Step 15. Only row 1, column 11 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 11, in indigo and in the ring around that square is now dead.
Step 16. All of the khaki 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: 6 squares come off.
Step 17. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: slate has one square left, so its queen takes row 5, column 8; 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 3; amber has one square left, so its queen takes row 2, column 2; blue has one square left, so its queen takes row 3, column 7; 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 6; sea green has one square left, so its queen takes row 10, 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 10, column 9.
The board after step 17 — 11 of 11 queens placed.
That completes the board: 11 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 11x11 Queens puzzle #6
Board #6 of the 11x11 bank carries a hard rating. Its 121 squares are shared between eleven colored regions, and the solution places eleven queens: one per row, one per column, one per region, none of them side by side or corner to corner.
Puzzle #6 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #6 are: indigo (3 squares), teal (3 squares), pale yellow (5 squares), mint (6 squares), amber (7 squares), red (7 squares), khaki (7 squares), orange (16 squares), blue (19 squares), slate (19 squares), sea green (29 squares). An even split would hand every color 11 squares, so the indigo region is running 8 squares under that average while the sea green region is the roomiest area on the grid with 29 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 indigo, with 3 squares, is under far more pressure than sea green with 29. A 26-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Red is the densest region on this grid at 88% of a 4×2 frame; sea green is the loosest, occupying only 59% of its 7×7 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 #6 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.
Indigo: 3 squares across rows 1–2, columns 10–11.
Teal covers 3 squares, spanning 2 rows and 2 columns (rows 10–11, columns 4–5).
Pale yellow occupies 5 squares within rows 5–7, columns 2–3.
Mint: 6 squares across rows 2–4, columns 9–11.
Amber covers 7 squares, spanning 3 rows and 3 columns (rows 1–3, columns 1–3).
Red occupies 7 squares within rows 3–6, columns 1–2.
Khaki: 7 squares across rows 7–9, columns 5–7.
Orange covers 16 squares, spanning 5 rows and 4 columns (rows 7–11, columns 1–4).
Blue occupies 19 squares within rows 1–4, columns 3–9.
Slate: 19 squares across rows 4–7, columns 4–11.
Sea green covers 29 squares, spanning 7 rows and 7 columns (rows 5–11, columns 5–11).
Where to start on this board
Begin where the choices are fewest. On this board that means indigo, which owns just 3 squares inside a 2×2 window running from row 1 to row 2 and column 10 to column 11. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
With 2 rows and 2 columns of room, indigo needs narrowing rather than solving outright. Test its 3 squares against the neighbours: a candidate that would strand another color is dead, and eliminating even two of them tightens the whole corner of the board around rows 1–2.
Work outward to the other flat regions next. Teal 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 #6
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 8, 9, 10 and 11 are the narrowest, touched by only 3 colors. Columns 1 and 8 are the tightest columns, with 3 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 10 rows. Row 7 is the crowded extreme at 5 colors — worth revisiting later, not now.
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.
Keep alternating between two views of the board. In the region view you ask which of the eleven colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 11x11 board unwinds.
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 #6 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers no single-line region at all, a 3-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 26. 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.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
Techniques that work well at 11x11
At 11x11 you are managing 121 squares, and holding the whole board in your head will not work. Split it into quadrants and solve locally. Most useful deductions on a grid this large are regional — a color pinched between two placed queens, or a band of rows that two regions must share. Trust the marks on the board over your memory.
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 #6, start hunting for it near row 1.
A non-spoiling distribution check: solved, #6 carries 3 queens on the perimeter against 8 queens inside, 1 of them cornered. 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 11x11 Queens puzzle #6?
Puzzle #6 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 3 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 11x11 puzzle #6 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #6, has been verified to have exactly one valid arrangement of eleven queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #6?
Start with the indigo region — the most constrained color on this board, with 3 squares spanning rows 1–2 and columns 10–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.