11x11 Queens Puzzle #40
Difficulty: medium. 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 15-step reasoning is written out below.
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Step-by-step solution for puzzle #40
What follows is a complete solve of puzzle #40, worked the way a person works it rather than the way a computer does. It runs to 15 deductions — 11 queen placements and 4 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 confined to one line.
Techniques this board needs: last square in a region, region confined to one line.
Show the full walkthrough (7 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: slate has one square left, so its queen takes row 5, column 5; mint has one square left, so its queen takes row 4, column 3; blue has one square left, so its queen takes row 3, column 10; teal has one square left, so its queen takes row 11, column 7; sea green has one square left, so its queen takes row 10, column 2. 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 2.
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: 4 squares come off.
Step 3. Amber cannot leave row 2 — every one of its open squares (2 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 2 squares in all.
Step 4. All of the red region's remaining squares (2 open squares) lie in column 9. 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 — 5 of 11 queens placed. Step 5. Only row 1, column 11 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 11, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 6. All of the pale yellow region's remaining squares (2 open squares) lie in row 7. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 7. 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: red has one square left, so its queen takes row 6, column 9; khaki has one square left, so its queen takes row 8, column 1; amber has one square left, so its queen takes row 2, column 4; pale yellow has one square left, so its queen takes row 7, column 6; orange has one square left, so its queen takes row 9, 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 9, column 8.
The board after step 7 — 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 #40
This is puzzle number 40 in the 11x11 collection, rated medium. 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 #40 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 #40 are: slate (1 square), teal (1 square), sea green (3 squares), blue (5 squares), indigo (6 squares), mint (7 squares), red (9 squares), khaki (12 squares), pale yellow (18 squares), amber (21 squares), orange (38 squares). An even split would hand every color 11 squares, so the slate region is running 10 squares under that average while the orange region is the roomiest area on the grid with 38 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Slate is working with 1 square while orange has 38 to spare, a gap of 37. Attack the cramped end and the roomy regions tend to resolve themselves.
Shape matters as much as size. The slate 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 8×5 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.
Slate covers 1 square, spanning 1 row and 1 column (rows 5–5, columns 5–5). It never leaves row 5, so its queen is locked to that row.
Teal occupies 1 square within rows 11–11, columns 7–7. It never leaves row 11, so its queen is locked to that row.
Sea green: 3 squares across rows 10–11, columns 1–2.
Blue covers 5 squares, spanning 3 rows and 2 columns (rows 3–5, columns 10–11).
Indigo occupies 6 squares within rows 1–3, columns 9–11.
Mint: 7 squares across rows 4–6, columns 1–4.
Red covers 9 squares, spanning 6 rows and 2 columns (rows 2–7, columns 8–9).
Khaki occupies 12 squares within rows 6–10, columns 1–3.
Pale yellow: 18 squares across rows 1–8, columns 4–8.
Amber covers 21 squares, spanning 4 rows and 7 columns (rows 1–4, columns 1–7).
Orange occupies 38 squares within rows 6–11, columns 3–11.
Where to start on this board
Open on the tightest color. Here that is slate, confined to 1 square between rows 5 and 5 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 #40.
Because slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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 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 #40
A quick way to find the next move is to count how many different colors touch each line. On this board, row 9 is the narrowest, touched by only 2 colors. Columns 6, 8, 9, 10 and 11 are the tightest columns, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 colors, so one of that handful must take the row and the rest are squeezed into the other 10 rows. Row 6 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
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.
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 #40 is rated medium
Ratings track how generous the opening is. On #40 you get four regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 37-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
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 9 on this board.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 8 queens in the middle, including 1 in a corner. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #40?
Puzzle #40 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #40 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #40, 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 #40?
Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.