11x11 Queens Puzzle #57
Difficulty: easy. 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 23-step reasoning is written out below.
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Step-by-step solution for puzzle #57
What follows is a complete solve of puzzle #57, worked the way a person works it rather than the way a computer does. It runs to 23 deductions — 11 queen placements and 12 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: region confined to one line, last square in a region, region shadow, line confined to one region, pairing argument on rows, pairing argument on columns.
Show the full walkthrough (18 steps) — contains the solution
Step 1. 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: 11 squares in all.
Step 2. The amber region is down to one open square, at row 2, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 1, the remaining amber squares, and the ring of neighbours around row 2, column 1.
Step 3. Only row 1, column 3 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 3, in indigo and in the ring around that square is now dead.
Step 4. All of the red 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: 6 squares come off.
Step 5. Pale yellow cannot leave column 11 — every one of its open squares (4 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 5 squares in all.
The board after step 5 — 2 of 11 queens placed. Step 6. The blue region has 3 squares left, spread across rows 3 and 4, columns 8 and 10. 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. Teal cannot leave row 11 — every one of its open squares (2 open squares) sits there. That row belongs to teal, so every square on it owned by another colour is dead: 7 squares in all.
Step 8. Orange is down to 3 candidates in rows 8 and 9, columns 4 and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
Step 9. Row 10 has only sea green squares left (5 open squares). The row needs a queen and only sea green can provide one, which pins that colour to this row and kills its squares elsewhere: 5 squares.
The board after step 9 — 2 of 11 queens placed. Step 10. Every open square left in row 9 (2 open squares) belongs to the orange region. Since that row must contain a queen, orange is the colour that supplies it — so orange squares anywhere else on the board are out, 1 square in all.
Step 11. Slate, red, pale yellow, and khaki are now boxed into rows 5–8 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 14 squares at once.
Step 12. Here is the pairing argument. The orange and teal regions have no open squares outside columns 4 and 5 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 8 squares come off in one stroke.
Step 13. Only row 8, column 6 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 6, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 14. The orange region is down to one open square, at row 9, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 4, the remaining orange squares, and the ring of neighbours around row 9, column 4.
The board after step 14 — 4 of 11 queens placed. Step 15. Only row 11, column 5 is still open in teal, and since each colour must take a queen, that settles it. Everything else in row 11, in column 5, in teal and in the ring around that square is now dead.
Step 16. Wherever mint ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 17. Blue cannot leave column 10 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 4 squares in all.
Step 18. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: slate has one square left, so its queen takes row 5, column 9; blue has one square left, so its queen takes row 3, column 10; mint has one square left, so its queen takes row 4, column 7; red has one square left, so its queen takes row 6, column 2; pale yellow has one square left, so its queen takes row 7, column 11; sea green has one square left, so its queen takes row 10, 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 10, column 8.
The board after step 18 — 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 #57
Puzzle #57 is one of the easy-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.
As with every puzzle in this collection, #57 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.
How the colors divide this board
From smallest to largest, the regions on puzzle #57 are: indigo (2 squares), amber (3 squares), pale yellow (4 squares), teal (4 squares), slate (7 squares), red (7 squares), khaki (7 squares), blue (13 squares), orange (15 squares), sea green (18 squares), mint (41 squares). An even split would hand every color 11 squares, so the indigo region is running 9 squares under that average while the mint region is the roomiest area on the grid with 41 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 indigo, with 2 squares, is under far more pressure than mint with 41. A 39-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 1×2 box that encloses it, while mint is the most scattered at 57% of its 8×9 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.
Indigo: 2 squares across rows 1–1, columns 2–3. It never leaves row 1, so its queen is locked to that row.
Amber covers 3 squares, spanning 2 rows and 2 columns (rows 1–2, columns 1–2).
Pale yellow occupies 4 squares within rows 5–8, columns 11–11. It sits wholly inside column 11, so its queen must be in that column.
Teal: 4 squares across rows 11–11, columns 2–5. It never leaves row 11, so its queen is locked to that row.
Slate covers 7 squares, spanning 4 rows and 3 columns (rows 5–8, columns 8–10).
Red occupies 7 squares within rows 5–9, columns 1–2.
Khaki: 7 squares across rows 6–8, columns 3–6.
Blue covers 13 squares, spanning 4 rows and 4 columns (rows 1–4, columns 8–11).
Orange occupies 15 squares within rows 7–11, columns 1–5.
Sea green: 18 squares across rows 9–11, columns 6–11.
Mint covers 41 squares, spanning 8 rows and 9 columns (rows 1–8, columns 1–9).
Where to start on this board
Indigo is the natural opening on #57: it is the smallest region on the grid at 2 squares, boxed into rows 1–1 and columns 2–3. Since one of those squares must hold a queen, that block of 1 row and 2 columns is the most contested part of the board.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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 amber 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 #57
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3, 4 and 10 are the narrowest, touched by only 2 colors. Column 7 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 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 10 rows. By contrast row 8 meets 6 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.
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 #57 is rated easy
Ratings track how generous the opening is. On #57 you get three regions pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 2 colors, and a 39-square gap between the largest and smallest regions. Those four numbers largely explain the easy 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.
A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.
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 #57, start hunting for it near row 3.
As a sanity check rather than a spoiler: the finished solution to #57 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
- 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 #57?
Puzzle #57 is rated easy. Its smallest color region holds 2 squares, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #57 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #57, 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 #57?
Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–1 and columns 2–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.