11x11 Queens Puzzle #81
Difficulty: easy. 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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #81
What follows is a complete solve of puzzle #81, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 11 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 region shadow.
Techniques this board needs: last square in a region, region shadow, region confined to one line.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Only row 2, column 1 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 1, the rest of the amber region and every square adjacent to it — diagonals included.
Step 2. Indigo is down to 3 candidates in rows 1, 3, and 4, columns 3 and 4, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. Red cannot leave column 11 — 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: 10 squares in all.
Step 4. Wherever khaki 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 — 1 of 11 queens placed. Step 5. Sea green is down to 3 candidates in rows 10 and 11, columns 2 and 3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about sea green itself.
Step 6. All of the teal region's remaining squares (2 open squares) lie in row 11. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.
Step 7. Only row 10, column 2 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 2, in sea green and in the ring around that square is now dead.
Step 8. Khaki is down to 3 candidates in rows 8 and 9, columns 6 and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
The board after step 8 — 2 of 11 queens placed. 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: orange has one square left, so its queen takes row 9, column 9; khaki has one square left, so its queen takes row 8, column 7; pale yellow has one square left, so its queen takes row 7, column 5; red has one square left, so its queen takes row 6, column 11; teal has one square left, so its queen takes row 11, column 6. 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 11, column 6.
Step 10. All of the blue region's remaining squares (2 open squares) lie in column 8. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 11. Slate cannot leave row 5 — every one of its open squares (2 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 2 squares in all.
Step 12. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: blue has one square left, so its queen takes row 3, column 8; indigo has one square left, so its queen takes row 1, column 3; mint has one square left, so its queen takes row 4, column 10; slate has one square left, so its queen takes row 5, column 4. 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 4.
The board after step 12 — 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 #81
Puzzle #81 is one of the easy-rated boards in the 11x11 set. Eleven colors divide its 121 squares, and you win by placing eleven queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
There is exactly one way to finish #81, 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 eleven regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #81 are: amber (1 square), red (2 squares), teal (3 squares), khaki (4 squares), sea green (6 squares), indigo (8 squares), orange (12 squares), blue (16 squares), mint (17 squares), slate (24 squares), pale yellow (28 squares). An even split would hand every color 11 squares, so the amber region is running 10 squares under that average while the pale yellow region is the roomiest area on the grid with 28 squares.
Every color takes one queen and one only, whatever its size — which is why the 27 square difference between amber and pale yellow 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.
Shape matters as much as size. The amber region is the most compact color here, filling 100% of the 1×1 box that encloses it, while blue is the most scattered at 44% of its 6×6 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
Here is the full footprint of every color on puzzle #81, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Amber: 1 square across rows 2–2, columns 1–1. It never leaves row 2, so its queen is locked to that row.
Red covers 2 squares, spanning 2 rows and 1 column (rows 6–7, columns 11–11). It sits wholly inside column 11, so its queen must be in that column.
Teal occupies 3 squares within rows 10–11, columns 6–7.
Khaki: 4 squares across rows 8–10, columns 6–7.
Sea green covers 6 squares, spanning 3 rows and 3 columns (rows 9–11, columns 1–3).
Indigo occupies 8 squares within rows 1–4, columns 1–4.
Orange: 12 squares across rows 8–11, columns 8–11.
Blue covers 16 squares, spanning 6 rows and 6 columns (rows 1–6, columns 4–9).
Mint occupies 17 squares within rows 1–6, columns 8–11.
Slate: 24 squares across rows 2–7, columns 1–6.
Pale yellow covers 28 squares, spanning 6 rows and 10 columns (rows 6–11, columns 1–10).
Where to start on this board
Amber is the natural opening on #81: it is the smallest region on the grid at 1 square, boxed into rows 2–2 and columns 1–1. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.
Because amber never leaves row 2, its queen must be in that row — and that instantly settles row 2 for everyone else. Sweep along it and X out every square that is not amber. 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. Red and teal 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 #81
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 5, 7 and 8 are the narrowest, touched by only 3 colors. Columns 5, 10 and 11 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 10 is the crowded extreme at 5 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 eleven colors looking for one with a single square left; sweep the eleven rows and eleven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 11x11 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 #81 is rated easy
The rating reflects how much the board concedes up front. Puzzle #81 has three regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 27 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
Stuck on #81 despite the easy 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 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.
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 #81 is row 1.
A non-spoiling distribution check: solved, #81 carries 4 queens on the perimeter against 7 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 11x11 Queens puzzle #81?
Puzzle #81 is rated easy. 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 11x11 puzzle #81 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #81, 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 #81?
Start with the amber region — the most constrained color on this board, with 1 square spanning rows 2–2 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.