11x11 Queens Puzzle #39
Difficulty: medium. 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 20-step reasoning is written out below.
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Step-by-step solution for puzzle #39
What follows is a complete solve of puzzle #39, worked the way a person works it rather than the way a computer does. It runs to 20 deductions — 11 queen placements and 9 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 rows, pairing argument on columns, last square in a row.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Only row 11, column 2 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 2, the rest of the teal region and every square adjacent to it — diagonals included.
Step 2. All of the indigo region's remaining squares (3 open squares) lie in row 1. 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 mint region has 4 squares left, spread across rows 2–4, columns 6 and 7. 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.
Step 4. Wherever sea green ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
The board after step 4 — 1 of 11 queens placed. Step 5. Orange and sea green are now boxed into rows 9 and 10 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 6 squares at once.
Step 6. All of the khaki region's remaining squares (3 open squares) lie in row 8. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.
Step 7. Wherever pale yellow ends up among its 6 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 8. Here is the pairing argument. The pale yellow and sea green 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: 6 squares come off in one stroke.
Step 9. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: indigo has one square left, so its queen takes row 1, column 9; blue has one square left, so its queen takes row 3, column 8; mint has one square left, so its queen takes row 4, column 6; red has one square left, so its queen takes row 6, 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 6, column 7.
The board after step 9 — 5 of 11 queens placed. Step 10. Pale yellow cannot leave column 11 — every one of its open squares (2 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 11. Row 10 has a single open square left, at column 10 in the sea green region. Every row needs a queen, so this one is forced. Row 10 and column 10 are now spoken for, along with the squares touching that queen.
Step 12. Khaki cannot leave row 8 — every one of its open squares (2 open squares) sits there. That row belongs to khaki, so every square on it owned by another colour is dead: 2 squares in all.
Step 13. 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: orange has one square left, so its queen takes row 9, column 3; amber has one square left, so its queen takes row 2, column 1; slate has one square left, so its queen takes row 5, column 4; pale yellow has one square left, so its queen takes row 7, column 11; khaki has one square left, so its queen takes row 8, 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 8, column 5.
The board after step 13 — 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 #39
11x11 puzzle #39 is rated medium. 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.
There is exactly one way to finish #39, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these eleven regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #39 are: teal (1 square), indigo (3 squares), mint (4 squares), khaki (6 squares), sea green (7 squares), pale yellow (9 squares), blue (11 squares), amber (15 squares), orange (17 squares), red (21 squares), slate (27 squares). An even split would hand every color 11 squares, so the teal region is running 10 squares under that average while the slate region is the roomiest area on the grid with 27 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Teal has only 1 place to put its queen; slate has 27. The spread between them on this board is 26 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×3 frame; slate is the loosest, occupying only 43% of its 9×7 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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.
Teal: 1 square across rows 11–11, columns 2–2. It never leaves row 11, so its queen is locked to that row.
Indigo covers 3 squares, spanning 1 row and 3 columns (rows 1–1, columns 9–11). It never leaves row 1, so its queen is locked to that row.
Mint occupies 4 squares within rows 2–4, columns 6–7.
Khaki: 6 squares across rows 8–10, columns 4–6.
Sea green covers 7 squares, spanning 3 rows and 4 columns (rows 9–11, columns 8–11).
Pale yellow occupies 9 squares within rows 3–9, columns 10–11.
Blue: 11 squares across rows 1–3, columns 6–11.
Amber covers 15 squares, spanning 6 rows and 4 columns (rows 1–6, columns 1–4).
Orange occupies 17 squares within rows 9–11, columns 1–8.
Red: 21 squares across rows 4–10, columns 6–10.
Slate covers 27 squares, spanning 9 rows and 7 columns (rows 1–9, columns 1–7).
Where to start on this board
The most constrained color is where a solve should begin, and on #39 that is teal with 1 square. It reaches from row 11 to row 11 and from column 2 to column 2 — 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.
Work outward to the other flat regions next. Indigo is no more than two rows tall, so its queen is 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 #39
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 7 and 11 are the narrowest, touched by only 3 colors. Columns 1, 3 and 5 are the tightest columns, with 3 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 10 rows. By contrast row 9 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
Working through the middle game
The middle of #39 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.
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.
Resist the plausible-looking queen. On #39 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #39 is rated medium
Ratings track how generous the opening is. On #39 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 26-square gap between the largest and smallest regions. Those four numbers largely explain the medium 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.
Stuck on #39 despite the medium rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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 #39 the place to look for one is around row 7.
As a sanity check rather than a spoiler: the finished solution to #39 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 #39?
Puzzle #39 is rated medium. 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 #39 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #39, 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 #39?
Start with the teal region — the most constrained color on this board, with 1 square spanning rows 11–11 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.