12x12 Queens Puzzle #75
Difficulty: hard. Place 12 queens on this 12x12 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 31-step reasoning is written out below.
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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 31 deductions — 12 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, region confined to one line, line confined to one region, pairing argument on rows, proof by contradiction, last square in a row, last square in a column.
Show the full walkthrough (23 steps) — contains the solution
Step 1. 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 2. Indigo is down to 3 candidates in rows 1 and 2, columns 4 and 5, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. The blue region has 5 squares left, spread across rows 1–3, 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 4. All of the red region's remaining squares (2 open squares) lie in column 12. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 11 squares come off.
Step 5. Rose cannot leave row 12 — every one of its open squares (3 open squares) sits there. That row belongs to rose, so every square on it owned by another colour is dead: 8 squares in all.
Step 6. Every open square left in row 11 (7 open squares) belongs to the teal region. Since that row must contain a queen, teal is the colour that supplies it — so teal squares anywhere else on the board are out, 9 squares in all.
The board after step 6 — 1 of 12 queens placed. Step 7. Indigo, amber, and blue are now boxed into rows 1–3 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 12 squares at once.
Step 8. All of the slate region's remaining squares (2 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 3 squares come off.
Step 9. Test row 2, column 5 for indigo: place a queen there, run the forced consequences through, and the khaki region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 10. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.
Step 11. Amber is down to 3 candidates in rows 2 and 3, columns 8 and 11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 12. The blue region has 3 squares left, spread across rows 2 and 3, columns 5 and 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.
The board after step 12 — 1 of 12 queens placed. Step 13. Test row 2, column 8 for amber: 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 amber queen went to row 3, column 8. 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 15. Only row 2, column 11 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 11, in amber and in the ring around that square is now dead.
Step 16. 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: 1 square come off.
Step 17. Test row 3, column 5 for blue: place a queen there, run the forced consequences through, and row 7 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 17 — 2 of 12 queens placed. Step 18. The blue region is down to one open square, at row 3, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 6, the remaining blue squares, and the ring of neighbours around row 3, column 6.
Step 19. Wherever khaki 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 20. Orange is down to 4 candidates in rows 5, 6, 7, and 9, columns 3 and 4, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about orange itself.
Step 21. Rose cannot leave row 12 — every one of its open squares (2 open squares) sits there. That row belongs to rose, so every square on it owned by another colour is dead: 1 square in all.
Step 22. Every open square left in row 8 (3 open squares) belongs to the khaki region. Since that row must contain a queen, khaki is the colour that supplies it — so khaki squares anywhere else on the board are out, 2 squares in all.
Step 23. 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: row 9 is down to one square, putting a queen on row 9, column 4; indigo has one square left, so its queen takes row 1, column 5; column 3 is down to one square, putting a queen on row 4, column 3; slate has one square left, so its queen takes row 5, column 1; red has one square left, so its queen takes row 6, column 12; column 10 is down to one square, putting a queen on row 11, column 10; column 9 is down to one square, putting a queen on row 7, column 9; khaki has one square left, so its queen takes row 8, column 7; rose has one square left, so its queen takes row 12, 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 12, column 8.
The board after step 23 — 12 of 12 queens placed.
That completes the board: 12 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 12x12 Queens puzzle #75
12x12 puzzle #75 is rated hard. Twelve colored regions cover its 144 squares, and your job is to seat twelve queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Puzzle #75 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #75 are: sea green (1 square), red (2 squares), indigo (3 squares), rose (3 squares), blue (6 squares), khaki (7 squares), amber (11 squares), slate (12 squares), mint (21 squares), pale yellow (23 squares), orange (25 squares), teal (30 squares). An even split would hand every color 12 squares, so the sea green region is running 11 squares under that average while the teal region is the roomiest area on the grid with 30 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Sea green has only 1 place to put its queen; teal has 30. The spread between them on this board is 29 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
A region's outline can constrain it more than its size does. Here red is the most solid, filling 100% of its 2×1 box, and mint the most diffuse at 50% of its 6×7 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Here is the full footprint of every color on puzzle #75, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Sea green: 1 square across rows 10–10, columns 2–2. It never leaves row 10, so its queen is locked to that row.
Red covers 2 squares, spanning 2 rows and 1 column (rows 5–6, columns 12–12). It sits wholly inside column 12, so its queen must be in that column.
Indigo occupies 3 squares within rows 1–2, columns 4–5.
Rose: 3 squares across rows 12–12, columns 6–8. It never leaves row 12, so its queen is locked to that row.
Blue covers 6 squares, spanning 3 rows and 3 columns (rows 1–3, columns 5–7).
Khaki occupies 7 squares within rows 7–9, columns 5–8.
Amber: 11 squares across rows 1–4, columns 8–12.
Slate covers 12 squares, spanning 5 rows and 4 columns (rows 1–5, columns 1–4).
Mint occupies 21 squares within rows 3–8, columns 1–7.
Pale yellow: 23 squares across rows 2–7, columns 7–12.
Orange covers 25 squares, spanning 8 rows and 6 columns (rows 5–12, columns 1–6).
Teal occupies 30 squares within rows 7–12, columns 4–12.
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 2 to column 2. 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. Rose, indigo and red 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, row 11 is the narrowest, touched by only 2 colors. Columns 1, 3, 9, 10 and 11 are the tightest columns, with 3 colors.
A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 11 rows. Row 7 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
The middle of #75 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 twelve queens to seat and twelve regions competing over 144 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 12x12 boards quickly and players who stall halfway.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #75 is rated hard
The rating reflects how much the board concedes up front. Puzzle #75 has four regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 29 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.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
Techniques that work well at 12x12
At 12x12 you are managing 144 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 11.
A non-spoiling distribution check: solved, #75 carries 4 queens on the perimeter against 8 queens inside, with the corners empty. 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 12x12 Queens puzzle #75?
Puzzle #75 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #75 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #75, has been verified to have exactly one valid arrangement of twelve 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 sea green region — the most constrained color on this board, with 1 square spanning rows 10–10 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.