11x11 Queens Puzzle #84
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 #84
What follows is a complete solve of puzzle #84, 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 pairing argument on columns.
Techniques this board needs: last square in a region, region shadow, region confined to one line, pairing argument on columns.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Only row 2, column 5 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 5, the rest of the amber region and every square adjacent to it — diagonals included.
Step 2. The teal region is down to one open square, at row 11, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 1, the remaining teal squares, and the ring of neighbours around row 11, column 1.
Step 3. The indigo region has 2 squares left, spread across rows 1 and 3, columns 2 and 3. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.
Step 4. All of the slate region's remaining squares (2 open squares) lie in column 11. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 9 squares come off.
Step 5. Only row 6, column 9 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 9, the rest of the red region and every square adjacent to it — diagonals included.
The board after step 5 — 3 of 11 queens placed. Step 6. The slate region is down to one open square, at row 5, column 11. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 11, the remaining slate squares, and the ring of neighbours around row 5, column 11.
Step 7. Mint cannot leave row 4 — every one of its open squares (2 open squares) sits there. That row belongs to mint, so every square on it owned by another colour is dead: 4 squares in all.
Step 8. The indigo region is down to one open square, at row 1, column 2. Every colour holds exactly one queen, so that square is forced. Row 1 and column 2 are now spoken for, along with the squares touching that queen.
Step 9. The sea green region has 2 squares left, spread across rows 7 and 10, columns 3 and 4. You do not need to know which of them takes the queen: every one of those options attacks the same 4 squares, so they can be crossed out now.
The board after step 9 — 5 of 11 queens placed. Step 10. Wherever pale yellow ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. Orange is down to 3 candidates in rows 9 and 10, columns 4 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
Step 12. Here is the pairing argument. The mint and sea green regions have no open squares outside columns 3 and 4 — 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: 2 squares come off in one stroke.
Step 13. Orange cannot leave column 6 — every one of its open squares (2 open squares) sits there. That column belongs to orange, so every square in it owned by another colour is dead: 3 squares in all.
Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: pale yellow has one square left, so its queen takes row 7, column 7; sea green has one square left, so its queen takes row 10, column 4; mint has one square left, so its queen takes row 4, column 3; orange has one square left, so its queen takes row 9, column 6; khaki has one square left, so its queen takes row 8, column 10; blue has one square left, so its queen takes row 3, column 8. 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 3, column 8.
The board after step 14 — 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 #84
This is puzzle number 84 in the 11x11 collection, rated easy. The grid holds 121 squares split between eleven colored regions, and a finished board carries eleven queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
Puzzle #84 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 #84 are: amber (1 square), teal (1 square), slate (2 squares), red (3 squares), indigo (7 squares), orange (9 squares), mint (16 squares), sea green (17 squares), pale yellow (18 squares), khaki (19 squares), blue (28 squares). An even split would hand every color 11 squares, so the amber region is running 10 squares under that average while the blue region is the roomiest area on the grid with 28 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 1 square while blue has 28 to spare, a gap of 27. Attack the cramped end and the roomy regions tend to resolve themselves.
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 pale yellow is the most scattered at 45% of its 5×8 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.
Amber: 1 square across rows 2–2, columns 5–5. It never leaves row 2, so its queen is locked to that row.
Teal covers 1 square, spanning 1 row and 1 column (rows 11–11, columns 1–1). It never leaves row 11, so its queen is locked to that row.
Slate occupies 2 squares within rows 5–6, columns 11–11. It sits wholly inside column 11, so its queen must be in that column.
Red: 3 squares across rows 5–6, columns 9–10.
Indigo covers 7 squares, spanning 3 rows and 3 columns (rows 1–3, columns 1–3).
Orange occupies 9 squares within rows 9–11, columns 3–8.
Mint: 16 squares across rows 1–6, columns 1–5.
Sea green covers 17 squares, spanning 6 rows and 5 columns (rows 6–11, columns 1–5).
Pale yellow occupies 18 squares within rows 4–8, columns 3–10.
Khaki: 19 squares across rows 7–11, columns 7–11.
Blue covers 28 squares, spanning 5 rows and 7 columns (rows 1–5, columns 5–11).
Where to start on this board
Open on the tightest color. Here that is amber, confined to 1 square between rows 2 and 2 and columns 5 and 5. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #84.
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. Teal, red 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 #84
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 3, 4, 7, 8, 9 and 10 are the narrowest, touched by only 3 colors. Columns 2, 6 and 11 are the tightest columns, with 3 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 10 rows. By contrast row 6 meets 5 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
With eleven queens to seat and eleven regions competing over 121 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 11x11 boards quickly and players who stall halfway.
Resist the plausible-looking queen. On #84 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 #84 is rated easy
Ratings track how generous the opening is. On #84 you get five regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 27-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.
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.
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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 1 on this board.
As a sanity check rather than a spoiler: the finished solution to #84 puts 3 queens on the outer border and 8 queens in the interior, 1 of them in an actual corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 #84?
Puzzle #84 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #84 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #84, 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 #84?
Start with the amber region — the most constrained color on this board, with 1 square spanning rows 2–2 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.