11x11 Queens Puzzle #75
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 19-step reasoning is written out below.
Previous puzzle · All 11x11 puzzles · Next puzzle
Step-by-step solution for puzzle #75
What follows is a complete solve of puzzle #75, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 11 queen placements and 8 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 shadow, pairing argument on rows, region confined to one line, line confined to one region, last square in a column.
Show the full walkthrough (13 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 sea green region is down to one open square, at row 10, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 1, the remaining sea green squares, and the ring of neighbours around row 10, column 1.
Step 3. The indigo region has 7 squares left, spread across rows 1–3, columns 8–11. 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 mint ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 4 — 2 of 11 queens placed. Step 5. Slate is down to 3 candidates in rows 4 and 5, columns 10 and 11, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about slate itself.
Step 6. The orange region has 3 squares left, spread across rows 8 and 9, 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 7. Indigo and amber are now boxed into rows 1 and 2 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 8. All of the blue region's remaining squares (2 open squares) lie in row 3. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.
Step 9. Only row 4, column 9 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 9, the rest of the mint region and every square adjacent to it — diagonals included.
The board after step 9 — 3 of 11 queens placed. Step 10. The slate region is down to one open square, at row 5, column 11. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 11, the remaining slate squares, and the ring of neighbours around row 5, column 11.
Step 11. Indigo is down to 2 candidates in rows 1 and 2, columns 8 and 10, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about indigo itself.
Step 12. Every open square left in column 2 (3 open squares) belongs to the khaki region. Since that column must contain a queen, khaki is the colour that supplies it — so khaki squares anywhere else on the board are out, 4 squares in all.
Step 13. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: column 3 is down to one square, putting a queen on row 6, column 3; khaki has one square left, so its queen takes row 8, column 2; orange has one square left, so its queen takes row 9, column 6; blue has one square left, so its queen takes row 3, column 4; amber has one square left, so its queen takes row 2, column 7; indigo has one square left, so its queen takes row 1, column 10; pale yellow has one square left, so its queen takes row 7, column 8. 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 7, column 8.
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 #75
11x11 puzzle #75 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, #75 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 #75 are: teal (1 square), slate (3 squares), sea green (4 squares), mint (5 squares), amber (6 squares), indigo (7 squares), blue (10 squares), orange (15 squares), red (18 squares), khaki (23 squares), pale yellow (29 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 29 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; pale yellow has 29. The spread between them on this board is 28 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. Teal is the densest region on this grid at 100% of a 1×1 frame; khaki is the loosest, occupying only 51% of its 9×5 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 5–5. It never leaves row 11, so its queen is locked to that row.
Slate covers 3 squares, spanning 2 rows and 2 columns (rows 4–5, columns 10–11).
Sea green occupies 4 squares within rows 10–11, columns 1–3.
Mint: 5 squares across rows 2–4, columns 8–10.
Amber covers 6 squares, spanning 2 rows and 5 columns (rows 1–2, columns 4–8).
Indigo occupies 7 squares within rows 1–3, columns 8–11.
Blue: 10 squares across rows 1–3, columns 1–6.
Orange covers 15 squares, spanning 4 rows and 7 columns (rows 8–11, columns 4–10).
Red occupies 18 squares within rows 3–8, columns 3–7.
Khaki: 23 squares across rows 2–10, columns 1–5.
Pale yellow covers 29 squares, spanning 8 rows and 6 columns (rows 4–11, columns 6–11).
Where to start on this board
The most constrained color is where a solve should begin, and on #75 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.
Work outward to the other flat regions next. Amber, slate and sea green 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 #75
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 6, 7 and 9 are the narrowest, touched by only 3 colors. Columns 1, 2 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 4 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
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.
Resist the plausible-looking queen. On #75 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 #75 is rated hard
Ratings track how generous the opening is. On #75 you get two regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 28-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
If #75 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 #75 the place to look for one is around row 1.
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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 11x11 Queens puzzle #75?
Puzzle #75 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #75 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #75, 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 #75?
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.