11x11 Queens Puzzle #87
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 14-step reasoning is written out below.
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Step-by-step solution for puzzle #87
What follows is a complete solve of puzzle #87, worked the way a person works it rather than the way a computer does. It runs to 14 deductions — 11 queen placements and 3 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 confined to one line, region shadow, last square in a column.
Show the full walkthrough (10 steps) — contains the solution
Step 1. The board opens by cascading. Each of these placements is forced the moment the one before it is marked, so the whole run is really a single sweep repeated: mint has one square left, so its queen takes row 4, column 3; indigo has one square left, so its queen takes row 1, column 4; orange has one square left, so its queen takes row 9, column 7; khaki has one square left, so its queen takes row 8, column 5; 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.
Step 2. All of the red 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: 3 squares come off.
Step 3. Teal cannot leave row 11 — every one of its open squares (2 open squares) sits there. That row belongs to teal, so every square on it owned by another colour is dead: 2 squares in all.
The board after step 3 — 5 of 11 queens placed. Step 4. The pale yellow region is down to one open square, at row 7, column 1. Every colour holds exactly one queen, so that square is forced. Row 7 and column 1 are now spoken for, along with the squares touching that queen.
Step 5. Only row 2, column 2 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 2, the rest of the amber region and every square adjacent to it — diagonals included.
Step 6. The red region is down to one open square, at row 6, column 11. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 11, the remaining red squares, and the ring of neighbours around row 6, column 11.
Step 7. Slate is down to 2 candidates in rows 3 and 5, columns 8 and 9, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about slate itself.
The board after step 7 — 8 of 11 queens placed. Step 8. Column 9 now has exactly one open square, row 5, column 9, in the slate region. Columns must each hold a queen, so that placement is forced. Everything else in row 5, in column 9, in slate and in the ring around that square is now dead.
Step 9. Only row 3, column 6 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 6, the rest of the blue region and every square adjacent to it — diagonals included.
Step 10. The teal region is down to one open square, at row 11, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 8, the remaining teal squares, and the ring of neighbours around row 11, column 8.
The board after step 10 — 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 #87
11x11 puzzle #87 is rated easy. Eleven colored regions cover its 121 squares, and your job is to seat eleven queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Puzzle #87 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 #87 are: mint (1 square), orange (1 square), indigo (2 squares), red (4 squares), sea green (5 squares), pale yellow (12 squares), khaki (12 squares), amber (16 squares), teal (16 squares), slate (18 squares), blue (34 squares). An even split would hand every color 11 squares, so the mint region is running 10 squares under that average while the blue region is the roomiest area on the grid with 34 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Mint is working with 1 square while blue has 34 to spare, a gap of 33. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×2 frame; blue is the loosest, occupying only 54% of its 7×9 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Here is the full footprint of every color on puzzle #87, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Mint: 1 square across rows 4–4, columns 3–3. It never leaves row 4, so its queen is locked to that row.
Orange covers 1 square, spanning 1 row and 1 column (rows 9–9, columns 7–7). It never leaves row 9, so its queen is locked to that row.
Indigo occupies 2 squares within rows 1–1, columns 3–4. It never leaves row 1, so its queen is locked to that row.
Red: 4 squares across rows 6–7, columns 10–11.
Sea green covers 5 squares, spanning 2 rows and 4 columns (rows 9–10, columns 8–11).
Pale yellow occupies 12 squares within rows 7–11, columns 1–3.
Khaki: 12 squares across rows 6–9, columns 3–7.
Amber covers 16 squares, spanning 8 rows and 3 columns (rows 1–8, columns 1–3).
Teal occupies 16 squares within rows 10–11, columns 3–11.
Slate: 18 squares across rows 3–8, columns 8–11.
Blue covers 34 squares, spanning 7 rows and 9 columns (rows 1–7, columns 3–11).
Where to start on this board
The most constrained color is where a solve should begin, and on #87 that is mint with 1 square. It reaches from row 4 to row 4 and from column 3 to column 3 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
Because mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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, orange, red and sea green 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 #87
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2 and 11 are the narrowest, touched by only 2 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 2 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 6 colors — worth revisiting later, not now.
Working through the middle game
The middle of #87 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.
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.
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 #87 is rated easy
The rating reflects how much the board concedes up front. Puzzle #87 has five regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 33 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.
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 2 on this board.
A non-spoiling distribution check: solved, #87 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 #87?
Puzzle #87 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #87 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #87, 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 #87?
Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.