11x11 Queens Puzzle #166
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 24-step reasoning is written out below.
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Step-by-step solution for puzzle #166
What follows is a complete solve of puzzle #166, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 11 queen placements and 13 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 confined to one line, region shadow, pairing argument on columns, line confined to one region, last square in a row.
Show the full walkthrough (19 steps) — contains the solution
Step 1. Only row 10, column 1 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 1, the rest of the sea green region and every square adjacent to it — diagonals included.
Step 2. All of the indigo region's remaining squares (2 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 12 squares come off.
Step 3. The mint region has 7 squares left, spread across rows 4–9, columns 10 and 11. 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 red ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Pale yellow is down to 5 candidates in rows 6 and 7, columns 3–6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
The board after step 5 — 1 of 11 queens placed. Step 6. Here is the pairing argument. The mint and teal regions have no open squares outside columns 10 and 11 — 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: 4 squares come off in one stroke.
Step 7. Only row 3, column 8 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 8, in blue and in the ring around that square is now dead.
Step 8. The indigo region is down to one open square, at row 1, column 9. Every colour holds exactly one queen, so that square is forced. Row 1 and column 9 are now spoken for, along with the squares touching that queen.
Step 9. Red cannot leave column 2 — 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: 4 squares in all.
Step 10. Every open square left in row 2 (4 open squares) belongs to the amber region. Since that row must contain a queen, amber is the colour that supplies it — so amber squares anywhere else on the board are out, 7 squares in all.
The board after step 10 — 3 of 11 queens placed. Step 11. Look along row 4: one square survives, row 4, column 10, which belongs to mint. A row cannot be left empty, so the queen goes there. That clears row 4, column 10, the rest of the mint region and every square adjacent to it — diagonals included.
Step 12. The teal region is down to one open square, at row 11, column 11. Every colour holds exactly one queen, so that square is forced. Row 11 and column 11 are now spoken for, along with the squares touching that queen.
Step 13. 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 14. Pale yellow is down to 4 candidates in rows 6 and 7, columns 4–6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
The board after step 14 — 5 of 11 queens placed. Step 15. The orange region has 4 squares left, spread across rows 8 and 9, columns 5–7. 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 16. Every open square left in row 5 (2 open squares) belongs to the slate region. Since that row must contain a queen, slate is the colour that supplies it — so slate squares anywhere else on the board are out, 2 squares in all.
Step 17. Slate cannot leave row 5 — every one of its open squares (2 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 1 square in all.
Step 18. The pale yellow region has 3 squares left, spread across rows 6 and 7, columns 4–6. 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 19. 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: row 7 is down to one square, putting a queen on row 7, column 6; slate has one square left, so its queen takes row 5, column 7; orange has one square left, so its queen takes row 9, column 5; khaki has one square left, so its queen takes row 8, column 3; amber has one square left, so its queen takes row 2, column 4; red has one square left, so its queen takes row 6, column 2. 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 6, column 2.
The board after step 19 — 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 #166
Board #166 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, #166 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 #166 are: sea green (1 square), indigo (2 squares), pale yellow (5 squares), mint (9 squares), teal (9 squares), blue (11 squares), red (13 squares), khaki (14 squares), orange (15 squares), slate (18 squares), amber (24 squares). An even split would hand every color 11 squares, so the sea green region is running 10 squares under that average while the amber region is the roomiest area on the grid with 24 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 sea green, with 1 square, is under far more pressure than amber with 24. A 23-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 2×1 frame; teal is the loosest, occupying only 50% of its 6×3 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Below is every color on #166 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.
Sea green covers 1 square, spanning 1 row and 1 column (rows 10–10, columns 1–1). It never leaves row 10, so its queen is locked to that row.
Indigo occupies 2 squares within rows 1–2, columns 9–9. It sits wholly inside column 9, so its queen must be in that column.
Pale yellow: 5 squares across rows 6–7, columns 3–6.
Mint covers 9 squares, spanning 6 rows and 3 columns (rows 4–9, columns 9–11).
Teal occupies 9 squares within rows 6–11, columns 9–11.
Blue: 11 squares across rows 1–4, columns 8–11.
Red covers 13 squares, spanning 8 rows and 3 columns (rows 1–8, columns 1–3).
Khaki occupies 14 squares within rows 7–11, columns 1–4.
Orange: 15 squares across rows 7–11, columns 4–9.
Slate covers 18 squares, spanning 7 rows and 4 columns (rows 4–10, columns 6–9).
Amber occupies 24 squares within rows 1–5, columns 2–7.
Where to start on this board
Begin where the choices are fewest. On this board that means sea green, which owns just 1 square inside a 1×1 window running from row 10 to row 10 and column 1 to column 1. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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. Indigo and pale yellow 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 #166
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 11 are the narrowest, touched by only 3 colors. Columns 1, 2, 5, 7, 8, 10 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 3 colors, so one of that handful must take the row and the rest are squeezed into the other 10 rows. Row 7 sits at the far end with 7 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
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 #166 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 3 colors, and a region-size spread of 23. 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 #166 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 #166, start hunting for it near row 3.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 8 queens in the middle, including 1 in a corner. 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 #166?
Puzzle #166 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #166 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #166, 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 #166?
Start with the sea green region — the most constrained color on this board, with 1 square spanning rows 10–10 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.