11x11 Queens Puzzle #291
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 #291
What follows is a complete solve of puzzle #291, 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 line confined to one region.
Techniques this board needs: last square in a region, region confined to one line, region shadow, line confined to one region, last square in a column, last square in a row.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Only row 11, column 5 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 5, the rest of the teal region and every square adjacent to it — diagonals included.
Step 2. 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.
Step 3. Only row 10, column 11 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 11, in sea green and in the ring around that square is now dead.
Step 4. All of the blue region's remaining squares (3 open squares) lie in column 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 6 squares come off.
The board after step 4 — 3 of 11 queens placed. Step 5. Slate cannot leave column 9 — 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: 8 squares in all.
Step 6. The indigo region has 4 squares left, spread across rows 1–3, columns 7 and 8. 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 7. Wherever khaki ends up among its 3 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
Step 8. Red is down to 5 candidates in rows 4–8, columns 6 and 8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 9. Column 1 has only mint squares left (8 open squares). The column needs a queen and only mint can provide one, which pins that colour to this column and kills its squares elsewhere: 6 squares.
The board after step 9 — 3 of 11 queens placed. Step 10. Every open square left in column 2 (3 open squares) belongs to the pale yellow region. Since that column must contain a queen, pale yellow is the colour that supplies it — so pale yellow squares anywhere else on the board are out, 5 squares in all.
Step 11. Scan column 3 and only row 2, column 3 remains — a amber square. Every column takes a queen, so it takes this one. Cross out the whole of row 2 and column 3, the remaining amber squares, and the ring of neighbours around row 2, column 3.
Step 12. The blue region is down to one open square, at row 3, column 10. Every colour holds exactly one queen, so that square is forced. Row 3 and column 10 are now spoken for, along with the squares touching that queen.
Step 13. Only row 5, column 9 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 9, the rest of the slate region and every square adjacent to it — diagonals included.
The board after step 13 — 6 of 11 queens placed. Step 14. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 1 square come off.
Step 15. Pale yellow cannot leave column 2 — every one of its open squares (3 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 1 square in all.
Step 16. All of the khaki 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: 2 squares come off.
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: indigo has one square left, so its queen takes row 1, column 8; row 7 is down to one square, putting a queen on row 7, column 2; mint has one square left, so its queen takes row 4, column 1; red has one square left, so its queen takes row 6, column 6; khaki has one square left, so its queen takes row 8, column 7. 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 8, column 7.
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 #291
11x11 puzzle #291 is rated hard. Eleven colored regions cover its 121 squares, and your job is to seat eleven queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Like every board on this site, #291 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.
How the colors divide this board
From smallest to largest, the regions on puzzle #291 are: teal (1 square), blue (5 squares), sea green (6 squares), orange (7 squares), indigo (9 squares), amber (11 squares), slate (11 squares), red (13 squares), mint (17 squares), khaki (18 squares), pale yellow (23 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 23 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 23 to spare, a gap of 22. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Teal is the densest region on this grid at 100% of a 1×1 frame; mint is the loosest, occupying only 43% of its 10×4 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a hard board.
Teal: 1 square across rows 11–11, columns 5–5. It never leaves row 11, so its queen is locked to that row.
Blue covers 5 squares, spanning 4 rows and 2 columns (rows 1–4, columns 10–11).
Sea green occupies 6 squares within rows 9–11, columns 9–11.
Orange: 7 squares across rows 9–11, columns 1–5.
Indigo covers 9 squares, spanning 5 rows and 4 columns (rows 1–5, columns 7–10).
Amber occupies 11 squares within rows 1–4, columns 3–7.
Slate: 11 squares across rows 3–8, columns 9–11.
Red covers 13 squares, spanning 5 rows and 5 columns (rows 4–8, columns 4–8).
Mint occupies 17 squares within rows 1–10, columns 1–4.
Khaki: 18 squares across rows 6–11, columns 6–10.
Pale yellow covers 23 squares, spanning 8 rows and 5 columns (rows 3–10, columns 2–6).
Where to start on this board
The most constrained color is where a solve should begin, and on #291 that is teal with 1 square. It reaches from row 11 to row 11 and from column 5 to column 5 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on 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.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: blue spans only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #291
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2 and 11 are the narrowest, touched by only 4 colors. Column 1 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 4 colors, then one of those regions has to own it, and every other region is competing for the remaining 10 rows. By contrast row 4 meets 7 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.
With eleven queens to seat and eleven regions competing over 121 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 11x11 boards quickly and players who stall halfway.
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 #291 is rated hard
The hard label comes out of four measurable things about #291: two of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 4 colors, and its largest and smallest regions differ by 22. Boards that score tight on those measures give up their first queen fast.
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.
If #291 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.
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 column 1 on this board.
As a sanity check rather than a spoiler: the finished solution to #291 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 #291?
Puzzle #291 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 4 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #291 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #291, 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 #291?
Start with the teal region — the most constrained color on this board, with 1 square spanning rows 11–11 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.