11x11 Queens Puzzle #85
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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #85
What follows is a complete solve of puzzle #85, 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 rows.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, line confined to one region, last square in a column, last square in a row.
Show the full walkthrough (11 steps) — contains the solution
Step 1. Only row 11, column 8 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 8, the rest of the teal region and every square adjacent to it — diagonals included.
Step 2. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 10 squares come off.
Step 3. Blue cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 8 squares in all.
Step 4. All of the pale yellow region's remaining squares (3 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.
The board after step 4 — 1 of 11 queens placed. Step 5. Only row 1, column 3 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 3, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 6. The amber region has 3 squares left, spread across rows 2 and 3, columns 5 and 6. 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.
Step 7. Wherever sea green ends up among its 3 squares, the same squares fall within reach of it — 4 squares in total, impossible regardless of how the colour resolves.
The board after step 7 — 2 of 11 queens placed. Step 8. Here is the pairing argument. The amber and blue regions have no open squares outside rows 2 and 3 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 9 squares come off in one stroke.
Step 9. The mint region has 3 squares left, spread across rows 4 and 5, columns 10 and 11. 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 10. Every open square left in column 9 (4 open squares) belongs to the khaki region. Since that column must contain a queen, khaki is the colour that supplies it — so khaki squares anywhere else on the board are out, 7 squares in all.
Step 11. 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: column 11 is down to one square, putting a queen on row 4, column 11; slate has one square left, so its queen takes row 5, column 4; column 10 is down to one square, putting a queen on row 10, column 10; row 9 is down to one square, putting a queen on row 9, column 5; amber has one square left, so its queen takes row 2, column 6; blue has one square left, so its queen takes row 3, column 1; red has one square left, so its queen takes row 6, column 7; pale yellow has one square left, so its queen takes row 7, column 2; khaki has one square left, so its queen takes row 8, 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 8, column 9.
The board after step 11 — 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 #85
Puzzle #85 is one of the hard-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.
As with every puzzle in this collection, #85 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 #85 are: teal (1 square), indigo (2 squares), blue (5 squares), red (9 squares), pale yellow (10 squares), sea green (10 squares), amber (12 squares), mint (12 squares), khaki (17 squares), slate (19 squares), orange (24 squares). An even split would hand every color 11 squares, so the teal region is running 10 squares under that average while the orange region is the roomiest area on the grid with 24 squares.
Every color takes one queen and one only, whatever its size — which is why the 23 square difference between teal and orange 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 indigo region is the most compact color here, filling 100% of the 1×2 box that encloses it, while slate is the most scattered at 45% of its 6×7 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
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 hard board.
Teal covers 1 square, spanning 1 row and 1 column (rows 11–11, columns 8–8). It never leaves row 11, so its queen is locked to that row.
Indigo occupies 2 squares within rows 1–1, columns 2–3. It never leaves row 1, so its queen is locked to that row.
Blue: 5 squares across rows 1–3, columns 1–3.
Red covers 9 squares, spanning 2 rows and 5 columns (rows 5–6, columns 4–8).
Pale yellow occupies 10 squares within rows 6–11, columns 1–2.
Sea green: 10 squares across rows 9–11, columns 6–11.
Amber covers 12 squares, spanning 4 rows and 5 columns (rows 1–4, columns 4–8).
Mint occupies 12 squares within rows 1–5, columns 9–11.
Khaki: 17 squares across rows 4–10, columns 8–11.
Slate covers 19 squares, spanning 6 rows and 7 columns (rows 2–7, columns 1–7).
Orange occupies 24 squares within rows 7–11, columns 2–8.
Where to start on this board
Teal is the natural opening on #85: it is the smallest region on the grid at 1 square, boxed into rows 11–11 and columns 8–8. 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 teal never leaves row 11, its queen must be in that row — and that instantly settles row 11 for everyone else. Sweep along it and X out every square that is not teal. 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 #85
A quick way to find the next move is to count how many different colors touch each line. On this board, row 8 is the narrowest, touched by only 3 colors. Columns 1, 9, 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 11 is the crowded extreme at 4 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 #85 is rated hard
The hard label comes out of four measurable things about #85: 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 23. Boards that score tight on those measures give up their first queen fast.
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.
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 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 #85 is row 8.
A non-spoiling distribution check: solved, #85 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 #85?
Puzzle #85 is rated hard. 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 #85 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #85, 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 #85?
Start with the teal region — the most constrained color on this board, with 1 square spanning rows 11–11 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.