11x11 Queens Puzzle #86
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 18-step reasoning is written out below.
Previous puzzle · All 11x11 puzzles · Next puzzle
Step-by-step solution for puzzle #86
What follows is a complete solve of puzzle #86, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 11 queen placements and 7 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 confined to one line, region shadow, pairing argument on columns, last square in a row.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Only row 1, column 7 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 7, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. All of the slate region's remaining squares (3 open squares) lie in column 11. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 8 squares come off.
Step 3. The amber region has 3 squares left, spread across rows 2 and 3, columns 9 and 10. 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 4. Wherever orange ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 4 — 1 of 11 queens placed. Step 5. Teal is down to 3 candidates in rows 10 and 11, columns 5 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about teal itself.
Step 6. Here is the pairing argument. The amber, red, and sea green regions have no open squares outside columns 8–10 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 4 squares come off in one stroke.
Step 7. Only row 9, column 6 is still open in orange, and since each colour must take a queen, that settles it. Everything else in row 9, in column 6, in orange and in the ring around that square is now dead.
Step 8. The teal region is down to one open square, at row 11, column 5. Every colour holds exactly one queen, so that square is forced. Row 11 and column 5 are now spoken for, along with the squares touching that queen.
The board after step 8 — 3 of 11 queens placed. Step 9. Blue cannot leave column 4 — 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: 7 squares in all.
Step 10. The pale yellow region is down to one open square, at row 7, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 3, the remaining pale yellow squares, and the ring of neighbours around row 7, column 3.
Step 11. Sea green cannot leave column 10 — every one of its open squares (2 open squares) sits there. That column belongs to sea green, so every square in it owned by another colour is dead: 4 squares in all.
Step 12. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 9; blue has one square left, so its queen takes row 3, column 4; row 8 is down to one square, putting a queen on row 8, column 1; mint has one square left, so its queen takes row 4, column 2; slate has one square left, so its queen takes row 5, column 11; red has one square left, so its queen takes row 6, column 8; sea green has one square left, so its queen takes row 10, column 10. 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 10.
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 #86
Board #86 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.
As with every puzzle in this collection, #86 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 #86 are: indigo (1 square), slate (3 squares), teal (3 squares), pale yellow (9 squares), blue (11 squares), orange (11 squares), red (14 squares), sea green (14 squares), amber (15 squares), mint (15 squares), khaki (25 squares). An even split would hand every color 11 squares, so the indigo region is running 10 squares under that average while the khaki region is the roomiest area on the grid with 25 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 1 place to put its queen; khaki has 25. The spread between them on this board is 24 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #86, packing 100% of its 1×1 bounding box, whereas orange straggles across its 6×4 box at just 46%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Here is the full footprint of every color on puzzle #86, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Indigo occupies 1 square within rows 1–1, columns 7–7. It never leaves row 1, so its queen is locked to that row.
Slate: 3 squares across rows 3–5, columns 11–11. It sits wholly inside column 11, so its queen must be in that column.
Teal covers 3 squares, spanning 2 rows and 2 columns (rows 10–11, columns 5–6).
Pale yellow occupies 9 squares within rows 7–9, columns 3–6.
Blue: 11 squares across rows 1–6, columns 3–6.
Orange covers 11 squares, spanning 6 rows and 4 columns (rows 6–11, columns 6–9).
Red occupies 14 squares within rows 3–8, columns 7–10.
Sea green: 14 squares across rows 6–11, columns 8–11.
Amber covers 15 squares, spanning 4 rows and 6 columns (rows 1–4, columns 6–11).
Mint occupies 15 squares within rows 1–5, columns 1–5.
Khaki: 25 squares across rows 5–11, columns 1–5.
Where to start on this board
Begin where the choices are fewest. On this board that means indigo, which owns just 1 square inside a 1×1 window running from row 1 to row 1 and column 7 to column 7. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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 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 #86
A quick way to find the next move is to count how many different colors touch each line. On this board, row 2 is the narrowest, touched by only 3 colors. Columns 1 and 2 are the tightest columns, with 2 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 8 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
The middle of #86 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.
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.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #86 is rated hard
The rating reflects how much the board concedes up front. Puzzle #86 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 24 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a hard board from the rest.
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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #86 the place to look for one is around column 1.
A non-spoiling distribution check: solved, #86 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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 11x11 Queens puzzle #86?
Puzzle #86 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 #86 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #86, 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 #86?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.