11x11 Queens Puzzle #71
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 30-step reasoning is written out below.
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Step-by-step solution for puzzle #71
What follows is a complete solve of puzzle #71, worked the way a person works it rather than the way a computer does. It runs to 30 deductions — 11 queen placements and 19 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 proof by contradiction.
Techniques this board needs: last square in a region, region shadow, pairing argument on columns, proof by contradiction, region confined to one line.
Show the full walkthrough (25 steps) — contains the solution
Step 1. Only row 11, column 4 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 4, the rest of the teal region and every square adjacent to it — diagonals included.
Step 2. Mint is down to 4 candidates in rows 2–4, columns 8 and 9, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about mint itself.
Step 3. The slate region has 6 squares left, spread across rows 2–6, columns 6 and 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 4. Wherever sea green 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.
Step 5. Blue and sea green are now boxed into columns 1 and 2 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 10 squares at once.
Step 6. Here is the pairing argument. The red and orange regions have no open squares outside columns 3 and 5 — 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: 7 squares come off in one stroke.
The board after step 6 — 1 of 11 queens placed. Step 7. Wherever indigo ends up among its 3 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 indigo and slate regions have no open squares outside columns 6 and 7 — 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: 10 squares come off in one stroke.
Step 9. The pale yellow region has 4 squares left, spread across rows 5–7, columns 8 and 9. 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 10. Here is the pairing argument. The mint and pale yellow regions have no open squares outside columns 8 and 9 — 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: 9 squares come off in one stroke.
Step 11. Test row 1, column 7 for indigo: place a queen there, run the forced consequences through, and the pale yellow region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 12. All of the indigo region's remaining squares (2 open squares) lie in column 6. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
Step 13. Test row 4, column 6 for indigo: place a queen there, run the forced consequences through, and row 10 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 13 — 1 of 11 queens placed. Step 14. The indigo region is down to one open square, at row 1, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 6, the remaining indigo squares, and the ring of neighbours around row 1, column 6.
Step 15. Amber cannot leave column 10 — every one of its open squares (4 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 16. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 10. Follow the forced moves that would create and the pale yellow region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 17. Test row 4, column 10 for amber: place a queen there, run the forced consequences through, and row 10 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 18. Now a short proof by contradiction. Suppose the amber queen went to row 5, column 10. Follow the forced moves that would create and column 9 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 19. Only row 2, column 10 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 10, in amber and in the ring around that square is now dead.
The board after step 19 — 3 of 11 queens placed. Step 20. The blue region is down to one open square, at row 3, column 1. Every colour holds exactly one queen, so that square is forced. Row 3 and column 1 are now spoken for, along with the squares touching that queen.
Step 21. Only row 10, column 2 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 2, the rest of the sea green region and every square adjacent to it — diagonals included.
Step 22. 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: 4 squares come off.
Step 23. Slate cannot leave column 7 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 1 square in all.
Step 24. The pale yellow region has 2 squares left, spread across rows 6 and 7, columns 8 and 9. 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 25. 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: slate has one square left, so its queen takes row 5, column 7; mint has one square left, so its queen takes row 4, column 9; red has one square left, so its queen takes row 6, column 3; pale yellow has one square left, so its queen takes row 7, column 8; 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 11. 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 11.
The board after step 25 — 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 #71
11x11 puzzle #71 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, #71 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 #71 are: teal (1 square), mint (4 squares), blue (5 squares), slate (7 squares), sea green (7 squares), indigo (8 squares), amber (10 squares), orange (14 squares), pale yellow (17 squares), red (24 squares), khaki (24 squares). An even split would hand every color 11 squares, so the teal region is running 10 squares under that average while the khaki region is the roomiest area on the grid with 24 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Teal is working with 1 square while khaki has 24 to spare, a gap of 23. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Teal is the densest region on this grid at 100% of a 1×1 frame; orange is the loosest, occupying only 40% of its 5×7 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Here is the full footprint of every color on puzzle #71, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Teal occupies 1 square within rows 11–11, columns 4–4. It never leaves row 11, so its queen is locked to that row.
Mint: 4 squares across rows 2–4, columns 8–9.
Blue covers 5 squares, spanning 3 rows and 2 columns (rows 1–3, columns 1–2).
Slate occupies 7 squares within rows 2–6, columns 6–8.
Sea green: 7 squares across rows 8–11, columns 1–3.
Indigo covers 8 squares, spanning 4 rows and 3 columns (rows 1–4, columns 5–7).
Amber occupies 10 squares within rows 1–5, columns 8–11.
Orange: 14 squares across rows 7–11, columns 2–8.
Pale yellow covers 17 squares, spanning 6 rows and 5 columns (rows 5–10, columns 5–9).
Red occupies 24 squares within rows 1–8, columns 1–5.
Khaki: 24 squares across rows 2–11, columns 8–11.
Where to start on this board
The most constrained color is where a solve should begin, and on #71 that is teal with 1 square. It reaches from row 11 to row 11 and from column 4 to column 4 — 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.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: blue and mint span only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #71
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, 9, 10 and 11 are the narrowest, touched by only 4 colors. Columns 10 and 11 are the tightest columns, with 2 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 4 colors has to be claimed by one of them, which pushes every remaining region into the other 10 rows. Row 3, meeting 7 colors, is the opposite case — the busiest line on #71 and the last one you should expect to crack.
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.
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.
A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.
Why puzzle #71 is rated hard
The rating reflects how much the board concedes up front. Puzzle #71 has two regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 4 colors, and the gap between its largest and smallest regions is 23 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a hard board from the rest.
If #71 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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around column 10 on this board.
Here is a shape check that gives nothing away: when #71 is solved, 4 queens sit on the outer edge of the grid and 7 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 #71?
Puzzle #71 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 4 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #71 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #71, 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 #71?
Start with the teal region — the most constrained color on this board, with 1 square spanning rows 11–11 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.