12x12 Queens Puzzle #209
Difficulty: medium. 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 23-step reasoning is written out below.
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Step-by-step solution for puzzle #209
What follows is a complete solve of puzzle #209, worked the way a person works it rather than the way a computer does. It runs to 23 deductions — 12 queen placements and 11 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 rows, proof by contradiction, last square in a row.
Show the full walkthrough (13 steps) — contains the solution
Step 1. The board opens by cascading. Each of these placements is forced the moment the one before it is marked, so the whole run is really a single sweep repeated: blue has one square left, so its queen takes row 3, column 11; slate has one square left, so its queen takes row 5, column 12; sea green has one square left, so its queen takes row 10, column 2; rose has one square left, so its queen takes row 12, column 1; teal has one square left, so its queen takes row 11, column 9. 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 11, column 9.
Step 2. The amber region has 2 squares left, spread across rows 2 and 4, columns 6 and 7. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.
Step 3. Wherever khaki 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 4. Here is the pairing argument. The red, pale yellow, khaki, and orange regions have no open squares outside rows 6–9 — 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: 3 squares come off in one stroke.
The board after step 4 — 5 of 12 queens placed. Step 5. Test row 1, column 3 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 6. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 5. Follow the forced moves that would create and row 2 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 7. Test row 1, column 6 for indigo: place a queen there, run the forced consequences through, and row 2 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 8. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 3. Follow the forced moves that would create and column 5 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 9. Test row 2, column 4 for indigo: place a queen there, run the forced consequences through, and column 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 9 — 5 of 12 queens placed. Step 10. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 5. Follow the forced moves that would create and column 4 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 11. The indigo region has 2 squares left, spread across rows 1 and 2, columns 4 and 8. 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 12. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 8. Follow the forced moves that would create and the red region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
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: indigo has one square left, so its queen takes row 1, column 4; row 2 is down to one square, putting a queen on row 2, column 6; khaki has one square left, so its queen takes row 8, column 5; pale yellow has one square left, so its queen takes row 7, column 3; mint has one square left, so its queen takes row 4, column 8; orange has one square left, so its queen takes row 9, column 10; red has one square left, so its queen takes row 6, 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 6, column 7.
The board after step 13 — 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 #209
Puzzle #209 is one of the medium-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 #209, 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 #209 are: blue (1 square), sea green (1 square), rose (2 squares), slate (6 squares), khaki (6 squares), teal (8 squares), red (9 squares), indigo (13 squares), amber (13 squares), mint (26 squares), orange (27 squares), pale yellow (32 squares). An even split would hand every color 12 squares, so the blue region is running 11 squares under that average while the pale yellow region is the roomiest area on the grid with 32 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Blue is working with 1 square while pale yellow has 32 to spare, a gap of 31. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Blue is the densest region on this grid at 100% of a 1×1 frame; mint is the loosest, occupying only 43% of its 6×10 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a medium board.
Blue occupies 1 square within rows 3–3, columns 11–11. It never leaves row 3, so its queen is locked to that row.
Sea green: 1 square across rows 10–10, columns 2–2. It never leaves row 10, so its queen is locked to that row.
Rose covers 2 squares, spanning 1 row and 2 columns (rows 12–12, columns 1–2). It never leaves row 12, so its queen is locked to that row.
Slate occupies 6 squares within rows 3–6, columns 11–12.
Khaki: 6 squares across rows 7–10, columns 5–6.
Teal covers 8 squares, spanning 2 rows and 7 columns (rows 11–12, columns 6–12).
Red occupies 9 squares within rows 5–9, columns 7–10.
Indigo: 13 squares across rows 1–2, columns 1–9.
Amber covers 13 squares, spanning 3 rows and 7 columns (rows 2–4, columns 1–7).
Mint occupies 26 squares within rows 1–6, columns 3–12.
Orange: 27 squares across rows 6–11, columns 7–12.
Pale yellow covers 32 squares, spanning 8 rows and 6 columns (rows 5–12, columns 1–6).
Where to start on this board
Open on the tightest color. Here that is blue, confined to 1 square between rows 3 and 3 and columns 11 and 11. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #209.
Because blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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. Sea green, rose, indigo and teal 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 #209
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Columns 1, 3, 4, 10 and 12 are the tightest columns, with 4 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 6 is the crowded extreme at 5 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 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.
Resist the plausible-looking queen. On #209 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 #209 is rated medium
The medium label comes out of four measurable things about #209: five of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 31. Boards that score tight on those measures give up their first queen fast.
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 #209 despite the medium 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.
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 row 1 on this board.
A non-spoiling distribution check: solved, #209 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 #209?
Puzzle #209 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #209 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #209, 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 #209?
Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.