11x11 Queens Puzzle #110
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 17-step reasoning is written out below.
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Step-by-step solution for puzzle #110
What follows is a complete solve of puzzle #110, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 11 queen placements and 6 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 shadow, region confined to one line, pairing argument on rows, pairing argument on columns.
Show the full walkthrough (10 steps) — contains the solution
Step 1. Only row 8, column 7 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 7, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 2. Indigo is down to 7 candidates in rows 1–3, columns 9–11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. The amber region has 3 squares left, spread across rows 1 and 2, columns 3 and 4. 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.
The board after step 3 — 1 of 11 queens placed. Step 4. All of the mint region's remaining squares (2 open squares) lie in row 4. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 12 squares come off.
Step 5. Indigo, amber, and blue are now boxed into rows 1–3 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 11 squares at once.
Step 6. Here is the pairing argument. The indigo, red, pale yellow, and sea green regions have no open squares outside columns 8–11 — 4 colours confined to 4 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 4 squares come off in one stroke.
Step 7. Only row 3, column 6 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 6, in blue and in the ring around that square is now dead.
The board after step 7 — 2 of 11 queens placed. Step 8. The mint region is down to one open square, at row 4, column 4. Every colour holds exactly one queen, so that square is forced. Row 4 and column 4 are now spoken for, along with the squares touching that queen.
Step 9. Amber cannot leave column 3 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 5 squares in all.
Step 10. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: slate has one square left, so its queen takes row 5, column 2; red has one square left, so its queen takes row 6, column 11; pale yellow has one square left, so its queen takes row 7, column 9; indigo has one square left, so its queen takes row 1, column 10; amber has one square left, so its queen takes row 2, column 3; sea green has one square left, so its queen takes row 10, column 8; orange has one square left, so its queen takes row 9, column 1; teal has one square left, so its queen takes row 11, column 5. 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 5.
The board after step 10 — 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 #110
Board #110 of the 11x11 bank carries a hard rating. Its 121 squares are shared between eleven colored regions, and the solution places eleven queens: one per row, one per column, one per region, none of them side by side or corner to corner.
Like every board on this site, #110 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 #110 are: khaki (1 square), mint (2 squares), amber (3 squares), indigo (7 squares), blue (8 squares), pale yellow (9 squares), orange (10 squares), red (11 squares), sea green (17 squares), teal (21 squares), slate (32 squares). An even split would hand every color 11 squares, so the khaki region is running 10 squares under that average while the slate region is the roomiest area on the grid with 32 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 khaki, with 1 square, is under far more pressure than slate with 32. A 31-square gap like this one rewards players who start where the choices are fewest.
Square count is only half the story — how a color is arranged counts too. Mint is the tidiest shape on #110, packing 100% of its 1×2 bounding box, whereas teal straggles across its 6×8 box at just 44%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Below is every color on #110 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Khaki occupies 1 square within rows 8–8, columns 7–7. It never leaves row 8, so its queen is locked to that row.
Mint: 2 squares across rows 4–4, columns 4–5. It never leaves row 4, so its queen is locked to that row.
Amber covers 3 squares, spanning 2 rows and 2 columns (rows 1–2, columns 3–4).
Indigo occupies 7 squares within rows 1–3, columns 9–11.
Blue: 8 squares across rows 1–4, columns 6–8.
Pale yellow covers 9 squares, spanning 3 rows and 4 columns (rows 5–7, columns 7–10).
Orange occupies 10 squares within rows 5–10, columns 1–2.
Red: 11 squares across rows 2–6, columns 7–11.
Sea green covers 17 squares, spanning 5 rows and 4 columns (rows 7–11, columns 8–11).
Teal occupies 21 squares within rows 6–11, columns 1–8.
Slate: 32 squares across rows 1–9, columns 1–6.
Where to start on this board
Begin where the choices are fewest. On this board that means khaki, which owns just 1 square inside a 1×1 window running from row 8 to row 8 and column 7 to column 7. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because khaki never leaves row 8, its queen must be in that row — and that instantly settles row 8 for everyone else. Sweep along it and X out every square that is not khaki. 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. Mint 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 #110
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, 2, 3, 5, 6 and 11 are the tightest columns, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 colors, so one of that handful must take the row and the rest are squeezed into the other 10 rows. Row 8 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
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 #110 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers three single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 31. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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 #110 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.
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 #110, start hunting for it near row 11.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 7 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #110?
Puzzle #110 is rated hard. Its smallest color region holds 1 square, 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 #110 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #110, 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 #110?
Start with the khaki region — the most constrained color on this board, with 1 square spanning rows 8–8 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.