12x12 Queens Puzzle #41
Difficulty: easy. 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 18-step reasoning is written out below.
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Step-by-step solution for puzzle #41
What follows is a complete solve of puzzle #41, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 12 queen placements and 6 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 region shadow.
Techniques this board needs: last square in a region, region shadow, region confined to one line.
Show the full walkthrough (13 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 mint region is down to one open square, at row 4, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 9, the remaining mint squares, and the ring of neighbours around row 4, column 9.
Step 3. The red region has 3 squares left, spread across rows 6 and 7, columns 2 and 3. 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 4. The pale yellow region is down to one open square, at row 7, column 1. Every colour holds exactly one queen, so that square is forced. Row 7 and column 1 are now spoken for, along with the squares touching that queen.
The board after step 4 — 3 of 12 queens placed. Step 5. Only row 6, column 3 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 3, the rest of the red region and every square adjacent to it — diagonals included.
Step 6. All of the slate region's remaining squares (2 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 7. Blue cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 4 squares in all.
Step 8. The khaki region is down to one open square, at row 8, column 4. Every colour holds exactly one queen, so that square is forced. Row 8 and column 4 are now spoken for, along with the squares touching that queen.
Step 9. The orange region has 5 squares left, spread across rows 9–11, 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.
The board after step 9 — 5 of 12 queens placed. Step 10. All of the teal region's remaining squares (3 open squares) lie in column 11. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 3 squares come off.
Step 11. Only row 5, column 7 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 7, in slate and in the ring around that square is now dead.
Step 12. The orange region has 2 squares left, spread across rows 9 and 11, columns 5 and 6. 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 13. 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: rose has one square left, so its queen takes row 12, column 5; orange has one square left, so its queen takes row 9, column 6; teal has one square left, so its queen takes row 11, column 11; sea green has one square left, so its queen takes row 10, column 8; amber has one square left, so its queen takes row 2, column 10; blue has one square left, so its queen takes row 3, column 2. 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 3, column 2.
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 #41
Puzzle #41 is one of the easy-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 #41, 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 #41 are: indigo (1 square), mint (1 square), pale yellow (2 squares), red (3 squares), sea green (10 squares), rose (10 squares), slate (16 squares), orange (16 squares), teal (19 squares), khaki (20 squares), blue (21 squares), amber (25 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the amber region is the roomiest area on the grid with 25 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 1 square while amber has 25 to spare, a gap of 24. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; rose is the loosest, occupying only 48% of its 3×7 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 easy board.
Indigo occupies 1 square within rows 1–1, columns 12–12. It never leaves row 1, so its queen is locked to that row.
Mint: 1 square across rows 4–4, columns 9–9. It never leaves row 4, so its queen is locked to that row.
Pale yellow covers 2 squares, spanning 2 rows and 1 column (rows 6–7, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.
Red occupies 3 squares within rows 6–7, columns 2–3.
Sea green: 10 squares across rows 9–12, columns 8–11.
Rose covers 10 squares, spanning 3 rows and 7 columns (rows 10–12, columns 1–7).
Slate occupies 16 squares within rows 4–7, columns 6–12.
Orange: 16 squares across rows 7–11, columns 4–8.
Teal covers 19 squares, spanning 7 rows and 5 columns (rows 6–12, columns 8–12).
Khaki occupies 20 squares within rows 5–11, columns 1–6.
Blue: 21 squares across rows 1–5, columns 1–6.
Amber covers 25 squares, spanning 4 rows and 9 columns (rows 1–4, columns 4–12).
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 #41.
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. Mint, pale yellow 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 #41
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2 and 3 are the narrowest, touched by only 2 colors. Columns 1, 2, 3, 7, 10, 11 and 12 are the tightest columns, with 4 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 11 rows. By contrast row 7 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
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 #41 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 #41 is rated easy
The easy label comes out of four measurable things about #41: 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 24. Boards that score tight on those measures give up their first queen fast.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
Stuck on #41 despite the easy 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 2 on this board.
As a sanity check rather than a spoiler: the finished solution to #41 puts 3 queens on the outer border and 9 queens in the interior, 1 of them in an actual corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 #41?
Puzzle #41 is rated easy. 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 #41 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #41, 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 #41?
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.