11x11 Queens Puzzle #232
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 16-step reasoning is written out below.
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Step-by-step solution for puzzle #232
What follows is a complete solve of puzzle #232, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 11 queen placements and 5 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: pairing argument on rows, last square in a region, region confined to one line, region shadow, last square in a row.
Show the full walkthrough (7 steps) — contains the solution
Step 1. Indigo, amber, blue, and mint are now boxed into rows 1–4 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 12 squares at once.
Step 2. The slate region is down to one open square, at row 5, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 1, the remaining slate squares, and the ring of neighbours around row 5, column 1.
Step 3. Amber cannot leave column 2 — every one of its open squares (3 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 6 squares in all.
Step 4. Wherever red ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
The board after step 4 — 1 of 11 queens placed. Step 5. Sea green and teal are now boxed into rows 10 and 11 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 7 squares at once.
Step 6. All of the orange 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: 8 squares come off.
Step 7. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: indigo has one square left, so its queen takes row 1, column 10; row 4 is down to one square, putting a queen on row 4, column 4; red has one square left, so its queen takes row 6, column 3; sea green has one square left, so its queen takes row 10, column 5; teal has one square left, so its queen takes row 11, column 7; pale yellow has one square left, so its queen takes row 7, column 8; blue has one square left, so its queen takes row 3, column 9; amber has one square left, so its queen takes row 2, column 2; khaki has one square left, so its queen takes row 8, column 6; orange has one square left, so its queen takes row 9, column 11. 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 11.
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 #232
This is puzzle number 232 in the 11x11 collection, rated hard. 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 #232 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 #232 are: indigo (5 squares), amber (5 squares), slate (5 squares), teal (6 squares), orange (7 squares), blue (9 squares), sea green (9 squares), mint (13 squares), pale yellow (14 squares), red (23 squares), khaki (25 squares). An even split would hand every color 11 squares, so the indigo region is running 6 squares under that average while the khaki region is the roomiest area on the grid with 25 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 5 squares while khaki has 25 to spare, a gap of 20. Attack the cramped end and the roomy regions tend to resolve themselves.
A region's outline can constrain it more than its size does. Here amber is the most solid, filling 83% of its 3×2 box, and red the most diffuse at 36% of its 8×8 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #232 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Indigo covers 5 squares, spanning 4 rows and 2 columns (rows 1–4, columns 10–11).
Amber occupies 5 squares within rows 1–3, columns 1–2.
Slate: 5 squares across rows 3–5, columns 1–3.
Teal covers 6 squares, spanning 2 rows and 4 columns (rows 10–11, columns 5–8).
Orange occupies 7 squares within rows 8–11, columns 8–11.
Blue: 9 squares across rows 1–3, columns 7–10.
Sea green covers 9 squares, spanning 3 rows and 5 columns (rows 9–11, columns 1–5).
Mint occupies 13 squares within rows 1–4, columns 3–6.
Pale yellow: 14 squares across rows 4–8, columns 7–11.
Red covers 23 squares, spanning 8 rows and 8 columns (rows 3–10, columns 1–8).
Khaki occupies 25 squares within rows 6–10, columns 3–10.
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 5 squares between rows 1 and 4 and columns 10 and 11. Its queen has nowhere else to go, which means those 4 rows and 2 columns carry more constraint than anywhere else on #232.
With 4 rows and 2 columns of room, indigo needs narrowing rather than solving outright. Test its 5 squares against the neighbours: a candidate that would strand another color is dead, and eliminating even two of them tightens the whole corner of the board around rows 1–4.
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 #232
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 5, 6, 7 and 11 are the narrowest, touched by only 3 colors. Column 11 is the tightest column, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 10 rows. Row 3, meeting 6 colors, is the opposite case — the busiest line on #232 and the last one you should expect to crack.
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.
A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.
Why puzzle #232 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers no single-line region at all, a 5-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 20. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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 5 on this board.
Here is a shape check that gives nothing away: when #232 is solved, 4 queens sit on the outer edge of the grid and 7 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 #232?
Puzzle #232 is rated hard. Its smallest color region holds 5 squares, its tightest row is touched by 3 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 11x11 puzzle #232 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #232, 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 #232?
Start with the indigo region — the most constrained color on this board, with 5 squares spanning rows 1–4 and columns 10–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.