11x11 Queens Puzzle #109

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 22-step reasoning is written out below.

Previous puzzle · All 11x11 puzzles · Next puzzle

Step-by-step solution for puzzle #109

What follows is a complete solve of puzzle #109, worked the way a person works it rather than the way a computer does. It runs to 22 deductions — 11 queen placements and 11 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 columns.

Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, line confined to one region, pairing argument on columns.

Show the full walkthrough (16 steps) — contains the solution

  1. Step 1. Only row 11, column 8 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 8, the rest of the teal region and every square adjacent to it — diagonals included.

  2. Step 2. The khaki region is down to one open square, at row 8, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 7, the remaining khaki squares, and the ring of neighbours around row 8, column 7.

  3. 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: 9 squares in all.

  4. 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.

  5. Step 5. Orange is down to 3 candidates in rows 9 and 10, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.

    Puzzle #109 after step 5: 2 of 11 queens placed
    The board after step 5 — 2 of 11 queens placed.

  6. Step 6. Here is the pairing argument. The orange and sea green regions have no open squares outside rows 9 and 10 — 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: 9 squares come off in one stroke.

  7. Step 7. Wherever red 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.

  8. Step 8. Pale yellow is down to 4 candidates in rows 6 and 7, columns 4–6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.

  9. Step 9. Column 3 has only slate squares left (6 open squares). The column needs a queen and only slate can provide one, which pins that colour to this column and kills its squares elsewhere: 14 squares.

  10. Step 10. Here is the pairing argument. The red and pale yellow regions have no open squares outside rows 6 and 7 — 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.

  11. Step 11. Indigo, blue, and red are now boxed into columns 9–11 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 2 squares at once.

    Puzzle #109 after step 11: 2 of 11 queens placed
    The board after step 11 — 2 of 11 queens placed.

  12. Step 12. The sea green region is down to one open square, at row 10, column 6. Every colour holds exactly one queen, so that square is forced. Row 10 and column 6 are now spoken for, along with the squares touching that queen.

  13. Step 13. Only row 9, column 1 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 1, the rest of the orange region and every square adjacent to it — diagonals included.

  14. Step 14. All of the amber region's remaining squares (2 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.

  15. Step 15. The mint region has 3 squares left, spread across rows 4 and 5, columns 4 and 5. 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.

  16. Step 16. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: slate has one square left, so its queen takes row 5, column 3; mint has one square left, so its queen takes row 4, column 5; pale yellow has one square left, so its queen takes row 7, column 4; red has one square left, so its queen takes row 6, column 9; indigo has one square left, so its queen takes row 1, column 10; blue has one square left, so its queen takes row 3, column 11; amber has one square left, so its queen takes row 2, 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 2, column 2.

    Puzzle #109 after step 16: 11 of 11 queens placed
    The board after step 16 — 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 #109

Puzzle #109 is one of the medium-rated boards in the 11x11 set. Eleven colors divide its 121 squares, and you win by placing eleven queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.

Puzzle #109 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 #109 are: teal (1 square), khaki (3 squares), indigo (4 squares), mint (5 squares), amber (6 squares), orange (7 squares), red (16 squares), sea green (16 squares), blue (19 squares), pale yellow (19 squares), slate (25 squares). An even split would hand every color 11 squares, so the teal region is running 10 squares under that average while the slate region is the roomiest area on the grid with 25 squares.

Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So teal, with 1 square, is under far more pressure than slate with 25. A 24-square gap like this one rewards players who start where the choices are fewest.

Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×4 box that encloses it, while red is the most scattered at 40% 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

Here is the full footprint of every color on puzzle #109, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

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.

Khaki occupies 3 squares within rows 7–8, columns 7–8.

Indigo: 4 squares across rows 1–1, columns 7–10. It never leaves row 1, so its queen is locked to that row.

Mint covers 5 squares, spanning 2 rows and 3 columns (rows 4–5, columns 4–6).

Amber occupies 6 squares within rows 1–3, columns 1–2.

Orange: 7 squares across rows 8–11, columns 1–2.

Red covers 16 squares, spanning 8 rows and 5 columns (rows 4–11, columns 7–11).

Sea green occupies 16 squares within rows 9–11, columns 3–10.

Blue: 19 squares across rows 1–6, columns 7–11.

Pale yellow covers 19 squares, spanning 6 rows and 6 columns (rows 5–10, columns 2–7).

Slate occupies 25 squares within rows 1–7, columns 1–6.

Where to start on this board

Teal is the natural opening on #109: it is the smallest region on the grid at 1 square, boxed into rows 11–11 and columns 8–8. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.

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, mint and khaki 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 #109

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2 and 3 are the narrowest, touched by only 3 colors. Column 11 is the tightest column, with 2 colors.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 10 rows. By contrast row 5 meets 5 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

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.

The whole middle game reduces to two sweeps. Sweep the eleven colors looking for one with a single square left; sweep the eleven rows and eleven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 11x11 solve from a stalled one.

The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.

Why puzzle #109 is rated medium

The rating reflects how much the board concedes up front. Puzzle #109 has three 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 24 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #109, start hunting for it near column 11.

As a sanity check rather than a spoiler: the finished solution to #109 puts 4 queens on the outer border and 7 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.

How to play Queens

  1. The board is an N×N grid divided into N colored regions.
  2. Place exactly one queen (♛) in each row, each column, and each colored region.
  3. No two queens may touch each other — not horizontally, vertically, or diagonally.
  4. 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.
  5. There is always exactly one valid solution. Use logic, never guessing, to find it.

Full rules · Strategy library · Puzzle statistics

Questions about this puzzle

How hard is 11x11 Queens puzzle #109?

Puzzle #109 is rated medium. 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 #109 have only one solution?

Yes. Every puzzle in the 11x11 collection, including #109, 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 #109?

Start with the teal region — the most constrained color on this board, with 1 square spanning rows 11–11 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.