11x11 Queens Puzzle #132
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 21-step reasoning is written out below.
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Step-by-step solution for puzzle #132
What follows is a complete solve of puzzle #132, worked the way a person works it rather than the way a computer does. It runs to 21 deductions — 11 queen placements and 10 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 proof by contradiction.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, proof by contradiction.
Show the full walkthrough (14 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: teal has one square left, so its queen takes row 11, column 2; orange has one square left, so its queen takes row 9, column 3; khaki has one square left, so its queen takes row 8, column 1; slate has one square left, so its queen takes row 5, column 4. 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 5, column 4.
Step 2. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 7 squares in all.
Step 3. 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 4. Wherever blue ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. 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: 3 squares come off in one stroke.
The board after step 5 — 4 of 11 queens placed. Step 6. Mint cannot leave row 4 — every one of its open squares (3 open squares) sits there. That row belongs to mint, so every square on it owned by another colour is dead: 3 squares in all.
Step 7. 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 8. Blue is down to 2 candidates in rows 2 and 3, columns 9 and 10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 9. The indigo region is down to one open square, at row 1, column 8. Every colour holds exactly one queen, so that square is forced. Row 1 and column 8 are now spoken for, along with the squares touching that queen.
The board after step 9 — 6 of 11 queens placed. Step 10. Mint cannot leave row 4 — every one of its open squares (2 open squares) sits there. That row belongs to mint, so every square on it owned by another colour is dead: 1 square in all.
Step 11. All of the amber region's remaining squares (2 open squares) lie in column 5. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 12. The pale yellow region has 2 squares left, spread across rows 7 and 10, columns 6 and 9. 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 13. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 5. Follow the forced moves that would create and the sea green region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 14. 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: amber has one square left, so its queen takes row 2, column 5; blue has one square left, so its queen takes row 3, column 9; pale yellow has one square left, so its queen takes row 7, column 6; mint has one square left, so its queen takes row 4, column 7; 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 14 — 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 #132
This is puzzle number 132 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.
There is exactly one way to finish #132, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these eleven regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #132 are: teal (1 square), indigo (2 squares), khaki (7 squares), blue (8 squares), slate (8 squares), orange (8 squares), mint (10 squares), red (10 squares), amber (20 squares), sea green (22 squares), pale yellow (25 squares). An even split would hand every color 11 squares, so the teal region is running 10 squares under that average while the pale yellow 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. Teal is working with 1 square while pale yellow has 25 to spare, a gap of 24. 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 indigo is the most solid, filling 100% of its 1×2 box, and red the most diffuse at 50% of its 4×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
Teal: 1 square across rows 11–11, columns 2–2. It never leaves row 11, so its queen is locked to that row.
Indigo covers 2 squares, spanning 1 row and 2 columns (rows 1–1, columns 8–9). It never leaves row 1, so its queen is locked to that row.
Khaki occupies 7 squares within rows 6–9, columns 1–3.
Blue: 8 squares across rows 1–3, columns 8–11.
Slate covers 8 squares, spanning 4 rows and 4 columns (rows 4–7, columns 1–4).
Orange occupies 8 squares within rows 9–11, columns 1–4.
Mint: 10 squares across rows 2–5, columns 4–8.
Red covers 10 squares, spanning 4 rows and 5 columns (rows 3–6, columns 7–11).
Amber occupies 20 squares within rows 1–5, columns 1–7.
Sea green: 22 squares across rows 6–11, columns 5–11.
Pale yellow covers 25 squares, spanning 7 rows and 6 columns (rows 5–11, columns 4–9).
Where to start on this board
Open on the tightest color. Here that is teal, confined to 1 square between rows 11 and 11 and columns 2 and 2. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #132.
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 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 #132
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 8 and 10 are the narrowest, touched by only 3 colors. Columns 10 and 11 are the tightest columns, 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 6, meeting 5 colors, is the opposite case — the busiest line on #132 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.
Two checks, run alternately, will carry you through the 121 squares of this grid. First: has any of the eleven regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 11x11 boards almost always run one of those checks and forget the other.
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 #132 is rated hard
Ratings track how generous the opening is. On #132 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 24-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
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.
Stuck on #132 despite the hard rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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 1 on this board.
Here is a shape check that gives nothing away: when #132 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 #132?
Puzzle #132 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 #132 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #132, 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 #132?
Start with the teal region — the most constrained color on this board, with 1 square spanning rows 11–11 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.