11x11 Queens Puzzle #45
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 27-step reasoning is written out below.
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Step-by-step solution for puzzle #45
What follows is a complete solve of puzzle #45, worked the way a person works it rather than the way a computer does. It runs to 27 deductions — 11 queen placements and 16 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 confined to one line, region shadow, pairing argument on rows, proof by contradiction, pairing argument on columns, last square in a column.
Show the full walkthrough (19 steps) — contains the solution
Step 1. Only row 11, column 1 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 1, the rest of the teal region and every square adjacent to it — diagonals included.
Step 2. 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: 12 squares come off.
Step 3. The mint region has 5 squares left, spread across rows 1, 2, 4, and 5, 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. All of the sea green region's remaining squares (3 open squares) lie in row 10. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.
Step 5. Pale yellow is down to 4 candidates in rows 7 and 9, columns 2–4, 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 indigo, amber, mint, and slate regions have no open squares outside rows 1, 2, 4, and 5 — 4 colours confined to 4 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 7 squares come off in one stroke.
Step 7. Red cannot leave row 6 — every one of its open squares (3 open squares) sits there. That row belongs to red, so every square on it owned by another colour is dead: 7 squares in all.
Step 8. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 2. Follow the forced moves that would create and the orange region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 9. Test row 1, column 3 for indigo: place a queen there, run the forced consequences through, and the orange region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 10. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 5. Follow the forced moves that would create and column 10 runs out of open squares entirely — impossible. So that square is ruled out for good.
The board after step 10 — 1 of 11 queens placed. Step 11. Test row 1, column 6 for indigo: place a queen there, run the forced consequences through, and column 10 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 12. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 4. Follow the forced moves that would create and the orange region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 13. Test row 2, column 5 for indigo: place a queen there, run the forced consequences through, and the orange region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 14. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 6. Follow the forced moves that would create and the sea green region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
The board after step 14 — 1 of 11 queens placed. Step 15. Indigo, blue, red, and pale yellow are now boxed into columns 2–5 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 9 squares at once.
Step 16. The sea green region is down to one open square, at row 10, column 6. Every colour holds exactly one queen, so that square is forced. Row 10 and column 6 are now spoken for, along with the squares touching that queen.
Step 17. Mint cannot leave column 7 — every one of its open squares (3 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 2 squares in all.
Step 18. All of the orange region's remaining squares (2 open squares) lie in row 9. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 7 squares come off.
Step 19. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: pale yellow has one square left, so its queen takes row 7, column 2; blue has one square left, so its queen takes row 3, column 3; red has one square left, so its queen takes row 6, column 5; indigo has one square left, so its queen takes row 1, column 4; column 11 is down to one square, putting a queen on row 8, column 11; orange has one square left, so its queen takes row 9, column 9; amber has one square left, so its queen takes row 2, column 8; mint has one square left, so its queen takes row 4, column 7; slate has one square left, so its queen takes row 5, column 10. 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 5, column 10.
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 #45
Puzzle #45 is one of the hard-rated boards in the 11x11 set. Eleven colors divide its 121 squares, and you win by placing eleven queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
There is exactly one way to finish #45, 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 #45 are: teal (1 square), blue (7 squares), mint (7 squares), slate (7 squares), red (7 squares), amber (8 squares), pale yellow (8 squares), sea green (9 squares), khaki (17 squares), indigo (18 squares), orange (32 squares). An even split would hand every color 11 squares, so the teal region is running 10 squares under that average while the orange 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 teal, with 1 square, is under far more pressure than orange 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. Teal is the tidiest shape on #45, packing 100% of its 1×1 bounding box, whereas orange straggles across its 7×10 box at just 46%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Here is the full footprint of every color on puzzle #45, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Teal: 1 square across rows 11–11, columns 1–1. It never leaves row 11, so its queen is locked to that row.
Blue covers 7 squares, spanning 5 rows and 3 columns (rows 2–6, columns 1–3).
Mint occupies 7 squares within rows 1–5, columns 6–8.
Slate: 7 squares across rows 2–5, columns 8–10.
Red covers 7 squares, spanning 2 rows and 4 columns (rows 5–6, columns 2–5).
Amber occupies 8 squares within rows 1–4, columns 8–11.
Pale yellow: 8 squares across rows 7–10, columns 1–4.
Sea green covers 9 squares, spanning 2 rows and 6 columns (rows 10–11, columns 1–6).
Khaki occupies 17 squares within rows 2–9, columns 8–11.
Indigo: 18 squares across rows 1–4, columns 1–6.
Orange covers 32 squares, spanning 7 rows and 10 columns (rows 5–11, columns 2–11).
Where to start on this board
Teal is the natural opening on #45: it is the smallest region on the grid at 1 square, boxed into rows 11–11 and columns 1–1. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of 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. Red 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 #45
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 7, 8, 9, 10 and 11 are the narrowest, touched by only 3 colors. Column 7 is the tightest column, with 2 colors.
A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 10 rows. Row 5 is the crowded extreme at 6 colors — worth revisiting later, not now.
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.
Two checks, run alternately, will carry you through the 121 squares of this grid. First: has any of the eleven regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 11x11 boards almost always run one of those checks and forget the other.
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 #45 is rated hard
The rating reflects how much the board concedes up front. Puzzle #45 has two regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 31 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.
Stuck on #45 despite the hard 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.
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 #45, start hunting for it near column 7.
A non-spoiling distribution check: solved, #45 carries 3 queens on the perimeter against 8 queens inside, 1 of them cornered. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.
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 #45?
Puzzle #45 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 #45 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #45, 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 #45?
Start with the teal region — the most constrained color on this board, with 1 square spanning rows 11–11 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.