11x11 Queens Puzzle #221
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 #221
What follows is a complete solve of puzzle #221, 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 pairing argument on columns.
Techniques this board needs: region confined to one line, region shadow, pairing argument on columns, last square in a region, last square in a column, pairing argument on rows.
Show the full walkthrough (17 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. All of the amber region's remaining squares (2 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.
Step 3. The blue region has 6 squares left, spread across rows 3–5, columns 1–4. 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 teal 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 4 — 0 of 11 queens placed. Step 5. Indigo, red, pale yellow, and teal are now boxed into columns 8–11 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 11 squares at once.
Step 6. All of the mint region's remaining squares (2 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 9 squares come off.
Step 7. Only row 2, column 5 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 5, in amber and in the ring around that square is now dead.
Step 8. All of the orange region's remaining squares (3 open squares) lie in column 6. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
Step 9. Only row 5, column 4 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 4, the rest of the slate region and every square adjacent to it — diagonals included.
The board after step 9 — 2 of 11 queens placed. Step 10. Column 8 now has exactly one open square, row 1, column 8, in the indigo region. Columns must each hold a queen, so that placement is forced. That clears row 1, column 8, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 11. Blue is down to 3 candidates in rows 3 and 4, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 12. Here is the pairing argument. The blue and mint regions have no open squares outside rows 3 and 4 — 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: 6 squares come off in one stroke.
Step 13. Only row 6, column 11 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 11, the rest of the red region and every square adjacent to it — diagonals included.
The board after step 13 — 4 of 11 queens placed. Step 14. The teal region is down to one open square, at row 11, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 10, the remaining teal squares, and the ring of neighbours around row 11, column 10.
Step 15. Only row 7, column 9 is still open in pale yellow, and since each colour must take a queen, that settles it. Everything else in row 7, in column 9, in pale yellow and in the ring around that square is now dead.
Step 16. Wherever khaki ends up among its 7 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 17. 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: sea green has one square left, so its queen takes row 10, column 2; blue has one square left, so its queen takes row 3, column 1; mint has one square left, so its queen takes row 4, column 7; khaki has one square left, so its queen takes row 8, column 3; orange has one square left, so its queen takes row 9, column 6. 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 6.
The board after step 17 — 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 #221
Puzzle #221 is one of the hard-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, #221 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 #221 are: indigo (2 squares), mint (4 squares), teal (4 squares), sea green (6 squares), amber (7 squares), pale yellow (10 squares), blue (12 squares), red (16 squares), slate (17 squares), khaki (21 squares), orange (22 squares). An even split would hand every color 11 squares, so the indigo region is running 9 squares under that average while the orange region is the roomiest area on the grid with 22 squares.
Every color takes one queen and one only, whatever its size — which is why the 20 square difference between indigo and orange is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
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 slate is the most scattered at 49% of its 7×5 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
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 occupies 2 squares within rows 1–1, columns 8–9. It never leaves row 1, so its queen is locked to that row.
Mint: 4 squares across rows 3–4, columns 7–8.
Teal covers 4 squares, spanning 3 rows and 2 columns (rows 9–11, columns 10–11).
Sea green occupies 6 squares within rows 9–11, columns 1–4.
Amber: 7 squares across rows 1–2, columns 4–8.
Pale yellow covers 10 squares, spanning 5 rows and 3 columns (rows 6–10, columns 9–11).
Blue occupies 12 squares within rows 1–5, columns 1–4.
Red: 16 squares across rows 1–6, columns 8–11.
Slate covers 17 squares, spanning 7 rows and 5 columns (rows 2–8, columns 3–7).
Khaki occupies 21 squares within rows 4–10, columns 1–4.
Orange: 22 squares across rows 6–11, columns 5–9.
Where to start on this board
Indigo is the natural opening on #221: it is the smallest region on the grid at 2 squares, boxed into rows 1–1 and columns 8–9. 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. Amber and mint 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 #221
A quick way to find the next move is to count how many different colors touch each line. On this board, row 11 is the narrowest, touched by only 3 colors. Columns 1, 2, 5, 6, 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 10, meeting 5 colors, is the opposite case — the busiest line on #221 and the last one you should expect to crack.
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 #221 is rated hard
Ratings track how generous the opening is. On #221 you get one region pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 3 colors, and a 20-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
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.
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.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #221 is row 11.
Here is a shape check that gives nothing away: when #221 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 #221?
Puzzle #221 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #221 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #221, 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 #221?
Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–1 and columns 8–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.