12x12 Queens Puzzle #261
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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #261
What follows is a complete solve of puzzle #261, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 12 queen placements and 7 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 columns.
Show the full walkthrough (10 steps) — contains the solution
Step 1. Only row 1, column 12 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 12, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The amber region is down to one open square, at row 2, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 2, the remaining amber squares, and the ring of neighbours around row 2, column 2.
Step 3. Blue cannot leave row 3 — every one of its open squares (2 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 6 squares in all.
The board after step 3 — 2 of 12 queens placed. Step 4. 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: 8 squares come off.
Step 5. Red is down to 4 candidates in rows 5 and 6, columns 8, 9, and 11, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about red itself.
Step 6. The orange region has 5 squares left, spread across rows 7–10, columns 8 and 9. 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. Wherever teal 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 7 — 2 of 12 queens placed. Step 8. Here is the pairing argument. The mint and rose regions have no open squares outside columns 1 and 3 — 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: 12 squares come off in one stroke.
Step 9. Red, khaki, orange, and teal are now boxed into columns 8–11 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 3 squares at once.
Step 10. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: blue has one square left, so its queen takes row 3, column 4; mint has one square left, so its queen takes row 4, column 1; rose has one square left, so its queen takes row 12, column 3; teal has one square left, so its queen takes row 11, column 9; orange has one square left, so its queen takes row 9, column 8; red has one square left, so its queen takes row 6, column 11; slate has one square left, so its queen takes row 5, column 5; khaki has one square left, so its queen takes row 8, column 10; sea green has one square left, so its queen takes row 10, column 6; pale yellow has one square left, so its queen takes row 7, 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 7, column 7.
The board after step 10 — 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 #261
Puzzle #261 is one of the hard-rated boards in the 12x12 set. Twelve colors divide its 144 squares, and you win by placing twelve 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 #261, 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 twelve regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #261 are: indigo (1 square), amber (3 squares), teal (4 squares), orange (5 squares), mint (6 squares), rose (7 squares), blue (15 squares), pale yellow (18 squares), slate (19 squares), khaki (20 squares), sea green (22 squares), red (24 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the red region is the roomiest area on the grid with 24 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 indigo, with 1 square, is under far more pressure than red with 24. A 23-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; slate is the loosest, occupying only 53% of its 6×6 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 #261, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Indigo: 1 square across rows 1–1, columns 12–12. It never leaves row 1, so its queen is locked to that row.
Amber covers 3 squares, spanning 2 rows and 2 columns (rows 1–2, columns 1–2).
Teal occupies 4 squares within rows 10–12, columns 9–10.
Orange: 5 squares across rows 7–10, columns 8–9.
Mint covers 6 squares, spanning 3 rows and 3 columns (rows 2–4, columns 1–3).
Rose occupies 7 squares within rows 10–12, columns 1–3.
Blue: 15 squares across rows 1–3, columns 3–8.
Pale yellow covers 18 squares, spanning 7 rows and 4 columns (rows 3–9, columns 5–8).
Slate occupies 19 squares within rows 4–9, columns 1–6.
Khaki: 20 squares across rows 5–12, columns 10–12.
Sea green covers 22 squares, spanning 5 rows and 7 columns (rows 8–12, columns 2–8).
Red occupies 24 squares within rows 1–6, columns 7–12.
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 12 and 12. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #261.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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 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 #261
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3, 4, 5, 6, 7, 11 and 12 are the narrowest, touched by only 4 colors. Column 11 is the tightest column, with 2 colors.
A line with few colors is a line close to resolving. Where only 4 colors appear, one of them owns the row outright and the others must fit into the remaining 11 rows. Row 10 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
The middle of #261 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.
Two checks, run alternately, will carry you through the 144 squares of this grid. First: has any of the twelve regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 12x12 boards almost always run one of those checks and forget the other.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #261 is rated hard
The rating reflects how much the board concedes up front. Puzzle #261 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.
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 #261 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 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.
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 #261, start hunting for it near column 11.
A non-spoiling distribution check: solved, #261 carries 3 queens on the perimeter against 9 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 12x12 Queens puzzle #261?
Puzzle #261 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 12x12 puzzle #261 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #261, 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 #261?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.