11x11 Queens Puzzle #180

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 36-step reasoning is written out below.

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Step-by-step solution for puzzle #180

What follows is a complete solve of puzzle #180, worked the way a person works it rather than the way a computer does. It runs to 36 deductions — 11 queen placements and 25 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, proof by contradiction, line confined to one region, last square in a row.

Show the full walkthrough (29 steps) — contains the solution

Step 1. Only row 2, column 1 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 1, the rest of the amber region and every square adjacent to it — diagonals included.
Step 2. 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.
Step 3. 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: 6 squares in all.
Step 4. Wherever mint 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.
Step 5. Slate cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 7 squares in all.
Step 6. The red region has 3 squares left, spread across rows 6 and 7, columns 10 and 11. 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 7. Wherever mint 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.
The board after step 7 — 2 of 11 queens placed.
Step 8. Orange is down to 5 candidates in rows 8 and 9, columns 5, 6, 7, and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
Step 9. The sea green region has 3 squares left, spread across rows 9 and 10, 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 10. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 4. Follow the forced moves that would create and the pale yellow region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 11. Test row 3, column 5 for blue: place a queen there, run the forced consequences through, and column 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 12. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 6. Follow the forced moves that would create and the orange region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 13. Test row 3, column 7 for blue: place a queen there, run the forced consequences through, and the orange region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 14. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 3. Follow the forced moves that would create and column 4 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 15. Test row 4, column 4 for blue: place a queen there, run the forced consequences through, and the khaki region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 15 — 2 of 11 queens placed.
Step 16. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 5. Follow the forced moves that would create and column 3 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 17. Test row 4, column 6 for blue: place a queen there, run the forced consequences through, and column 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 18. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 7. Follow the forced moves that would create and column 3 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 19. Test row 5, column 4 for blue: place a queen there, run the forced consequences through, and row 4 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 20. Now a short proof by contradiction. Suppose the blue queen went to row 6, column 4. Follow the forced moves that would create and row 4 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 21. Column 4 has only pale yellow squares left (3 open squares). The column needs a queen and only pale yellow can provide one, which pins that colour to this column and kills its squares elsewhere: 9 squares.
Step 22. All of the pale yellow region's remaining squares (3 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
The board after step 22 — 2 of 11 queens placed.
Step 23. Column 3 has only blue squares left (2 open squares). The column needs a queen and only blue can provide one, which pins that colour to this column and kills its squares elsewhere: 1 square.
Step 24. Every open square left in column 5 (2 open squares) belongs to the sea green region. Since that column must contain a queen, sea green is the colour that supplies it — so sea green squares anywhere else on the board are out, 1 square in all.
Step 25. Sea green cannot leave column 5 — 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: 1 square in all.
Step 26. All of the pale yellow region's remaining squares (2 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
Step 27. Only row 3, column 3 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 3, in blue and in the ring around that square is now dead.
Step 28. Wherever mint ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 29. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: row 4 is down to one square, putting a queen on row 4, column 11; red has one square left, so its queen takes row 6, column 10; slate has one square left, so its queen takes row 5, column 2; khaki has one square left, so its queen takes row 8, column 9; pale yellow has one square left, so its queen takes row 7, column 4; orange has one square left, so its queen takes row 9, column 7; indigo has one square left, so its queen takes row 1, column 6; sea green has one square left, so its queen takes row 10, column 5. 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 5.
The board after step 29 — 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 #180

This is puzzle number 180 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 #180 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 #180 are: amber (1 square), teal (1 square), red (3 squares), indigo (5 squares), slate (6 squares), orange (7 squares), mint (12 squares), pale yellow (13 squares), sea green (18 squares), khaki (24 squares), blue (31 squares). An even split would hand every color 11 squares, so the amber region is running 10 squares under that average while the blue region is the roomiest area on the grid with 31 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 1 square while blue has 31 to spare, a gap of 30. 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 100% of its 1×1 box, and pale yellow the most diffuse at 43% of its 6×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

Region by region

Here is the full footprint of every color on puzzle #180, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Amber: 1 square across rows 2–2, columns 1–1. It never leaves row 2, so its queen is locked to that row.

Teal covers 1 square, spanning 1 row and 1 column (rows 11–11, columns 8–8). It never leaves row 11, so its queen is locked to that row.

Red occupies 3 squares within rows 6–7, columns 10–11.

Indigo: 5 squares across rows 1–2, columns 6–8.

Slate covers 6 squares, spanning 4 rows and 2 columns (rows 4–7, columns 1–2).

Orange occupies 7 squares within rows 8–10, columns 5–9.

Mint: 12 squares across rows 1–5, columns 8–11.

Pale yellow covers 13 squares, spanning 6 rows and 5 columns (rows 5–10, columns 3–7).

Sea green occupies 18 squares within rows 8–11, columns 1–7.

Khaki: 24 squares across rows 3–11, columns 7–11.

Blue covers 31 squares, spanning 8 rows and 7 columns (rows 1–8, columns 1–7).

Where to start on this board

Open on the tightest color. Here that is amber, confined to 1 square between rows 2 and 2 and columns 1 and 1. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #180.

Because amber never leaves row 2, its queen must be in that row — and that instantly settles row 2 for everyone else. Sweep along it and X out every square that is not amber. 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, indigo and red 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 #180

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 3 and 11 are the narrowest, touched by only 3 colors. Columns 2, 3, 4, 9, 10 and 11 are the tightest columns, with 3 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 #180 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.

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 #180 is rated medium

The rating reflects how much the board concedes up front. Puzzle #180 has four 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 30 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 medium board from the rest.

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 1 on this board.

A non-spoiling distribution check: solved, #180 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 #180?

Puzzle #180 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.

Does 11x11 puzzle #180 have only one solution?

Yes. Every puzzle in the 11x11 collection, including #180, 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 #180?

Start with the amber region — the most constrained color on this board, with 1 square spanning rows 2–2 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.