11x11 Queens Puzzle #31
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 16-step reasoning is written out below.
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Step-by-step solution for puzzle #31
What follows is a complete solve of puzzle #31, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 11 queen placements and 5 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 rows.
Techniques this board needs: last square in a region, region confined to one line, pairing argument on rows, line confined to one region, last square in a column.
Show the full walkthrough (11 steps) — contains the solution
Step 1. Only row 1, column 11 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 11, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. 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. Cross out the whole of row 3 and column 10, the remaining blue squares, and the ring of neighbours around row 3, column 10.
Step 3. Red cannot leave column 5 — every one of its open squares (2 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 11 squares in all.
Step 4. 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. Row 9 and column 4 are now spoken for, along with the squares touching that queen.
The board after step 4 — 3 of 11 queens placed. Step 5. Teal cannot leave row 11 — every one of its open squares (3 open squares) sits there. That row belongs to teal, so every square on it owned by another colour is dead: 5 squares in all.
Step 6. Here is the pairing argument. The amber, mint, and slate regions have no open squares outside rows 2, 4, and 5 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 5 squares come off in one stroke.
Step 7. Only row 6, column 5 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 5, in red and in the ring around that square is now dead.
The board after step 7 — 4 of 11 queens placed. Step 8. The pale yellow region is down to one open square, at row 7, column 7. Every colour holds exactly one queen, so that square is forced. Row 7 and column 7 are now spoken for, along with the squares touching that queen.
Step 9. Column 9 has only sea green squares left (2 open squares). The column needs a queen and only sea green can provide one, which pins that colour to this column and kills its squares elsewhere: 2 squares.
Step 10. Every open square left in column 8 (3 open squares) belongs to the slate region. Since that column must contain a queen, slate is the colour that supplies it — so slate squares anywhere else on the board are out, 1 square in all.
Step 11. 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: column 6 is down to one square, putting a queen on row 2, column 6; mint has one square left, so its queen takes row 4, column 1; slate has one square left, so its queen takes row 5, column 8; khaki has one square left, so its queen takes row 8, column 2; sea green has one square left, so its queen takes row 10, column 9; teal has one square left, so its queen takes row 11, column 3. 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 11, column 3.
The board after step 11 — 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 #31
11x11 puzzle #31 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, #31 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 #31 are: indigo (1 square), red (2 squares), orange (4 squares), teal (4 squares), blue (6 squares), pale yellow (8 squares), mint (9 squares), amber (15 squares), slate (16 squares), khaki (26 squares), sea green (30 squares). An even split would hand every color 11 squares, so the indigo region is running 10 squares under that average while the sea green region is the roomiest area on the grid with 30 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 1 place to put its queen; sea green has 30. The spread between them on this board is 29 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and amber the most diffuse at 50% of its 5×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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 covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 11–11). It never leaves row 1, so its queen is locked to that row.
Red occupies 2 squares within rows 5–6, columns 5–5. It sits wholly inside column 5, so its queen must be in that column.
Orange: 4 squares across rows 8–10, columns 4–5.
Teal covers 4 squares, spanning 1 row and 4 columns (rows 11–11, columns 1–4). It never leaves row 11, so its queen is locked to that row.
Blue occupies 6 squares within rows 2–5, columns 10–11.
Pale yellow: 8 squares across rows 5–7, columns 6–9.
Mint covers 9 squares, spanning 4 rows and 3 columns (rows 1–4, columns 1–3).
Amber occupies 15 squares within rows 1–5, columns 2–7.
Slate: 16 squares across rows 1–5, columns 6–10.
Khaki covers 26 squares, spanning 7 rows and 6 columns (rows 4–10, columns 1–6).
Sea green occupies 30 squares within rows 6–11, columns 5–11.
Where to start on this board
The most constrained color is where a solve should begin, and on #31 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 11 to column 11 — 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 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 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 #31
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 2 colors. Columns 1, 8, 9, 10 and 11 are the tightest columns, with 3 colors.
A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 10 rows. Row 5 is the crowded extreme at 6 colors — worth revisiting later, not now.
Working through the middle game
The middle of #31 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.
Keep alternating between two views of the board. In the region view you ask which of the eleven colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 11x11 board unwinds.
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 #31 is rated hard
Ratings track how generous the opening is. On #31 you get four regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 29-square gap between the largest and smallest regions. Those four numbers largely explain the hard 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.
If #31 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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #31 the place to look for one is around row 11.
A non-spoiling distribution check: solved, #31 carries 3 queens on the perimeter against 8 queens inside, 1 of them cornered. 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 #31?
Puzzle #31 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #31 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #31, 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 #31?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.