12x12 Queens Puzzle #46
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 20-step reasoning is written out below.
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Step-by-step solution for puzzle #46
What follows is a complete solve of puzzle #46, worked the way a person works it rather than the way a computer does. It runs to 20 deductions — 12 queen placements and 8 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 rows.
Techniques this board needs: last square in a region, region shadow, region confined to one line, pairing argument on rows, last square in a column.
Show the full walkthrough (14 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: slate has one square left, so its queen takes row 5, column 7; pale yellow has one square left, so its queen takes row 7, column 12; 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 5. 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 5.
Step 2. Indigo is down to 4 candidates in rows 1–3, columns 3 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. The amber region has 3 squares left, spread across rows 1 and 2, columns 6 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 4. Wherever khaki ends up among its 6 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. The sea green region is down to one open square, at row 10, column 2. Every colour holds exactly one queen, so that square is forced. Row 10 and column 2 are now spoken for, along with the squares touching that queen.
The board after step 5 — 5 of 12 queens placed. Step 6. Khaki cannot leave column 4 — every one of its open squares (2 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 5 squares in all.
Step 7. The indigo region is down to one open square, at row 1, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 3, the remaining indigo squares, and the ring of neighbours around row 1, column 3.
Step 8. Only row 2, column 8 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 8, in amber and in the ring around that square is now dead.
Step 9. 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. Row 3 and column 6 are now spoken for, along with the squares touching that queen.
The board after step 9 — 8 of 12 queens placed. Step 10. Wherever mint ends up among its 2 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. Red is down to 4 candidates in rows 4, 6, 8, and 9, columns 9–11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 12. The orange region has 2 squares left, spread across rows 8 and 9, columns 9 and 10. 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 13. Here is the pairing argument. The khaki and orange regions have no open squares outside rows 8 and 9 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 1 square come off in one stroke.
Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: column 11 is down to one square, putting a queen on row 4, column 11; red has one square left, so its queen takes row 6, column 9; orange has one square left, so its queen takes row 9, column 10; khaki has one square left, so its queen takes row 8, column 4. 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 8, column 4.
The board after step 14 — 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 #46
Board #46 of the 12x12 bank carries a easy rating. Its 144 squares are shared between twelve colored regions, and the solution places twelve queens: one per row, one per column, one per region, none of them side by side or corner to corner.
As with every puzzle in this collection, #46 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.
How the colors divide this board
From smallest to largest, the regions on puzzle #46 are: slate (1 square), pale yellow (1 square), rose (1 square), indigo (5 squares), amber (5 squares), khaki (9 squares), teal (11 squares), red (15 squares), sea green (15 squares), mint (19 squares), orange (26 squares), blue (36 squares). An even split would hand every color 12 squares, so the slate region is running 11 squares under that average while the blue region is the roomiest area on the grid with 36 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 slate, with 1 square, is under far more pressure than blue with 36. A 35-square gap like this one rewards players who start where the choices are fewest.
A region's outline can constrain it more than its size does. Here slate is the most solid, filling 100% of its 1×1 box, and red the most diffuse at 50% of its 6×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Slate covers 1 square, spanning 1 row and 1 column (rows 5–5, columns 7–7). It never leaves row 5, so its queen is locked to that row.
Pale yellow occupies 1 square within rows 7–7, columns 12–12. It never leaves row 7, so its queen is locked to that row.
Rose: 1 square across rows 12–12, columns 1–1. It never leaves row 12, so its queen is locked to that row.
Indigo covers 5 squares, spanning 3 rows and 3 columns (rows 1–3, columns 3–5).
Amber occupies 5 squares within rows 1–2, columns 6–8.
Khaki: 9 squares across rows 7–10, columns 2–5.
Teal covers 11 squares, spanning 2 rows and 9 columns (rows 11–12, columns 4–12).
Red occupies 15 squares within rows 4–9, columns 7–11.
Sea green: 15 squares across rows 7–12, columns 1–4.
Mint covers 19 squares, spanning 6 rows and 4 columns (rows 1–6, columns 9–12).
Orange occupies 26 squares within rows 8–11, columns 5–12.
Blue: 36 squares across rows 1–7, columns 1–8.
Where to start on this board
Slate is the natural opening on #46: it is the smallest region on the grid at 1 square, boxed into rows 5–5 and columns 7–7. 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 slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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. Pale yellow, rose, amber 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 #46
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3, 4, 6, 10, 11 and 12 are the narrowest, touched by only 3 colors. Columns 1 and 2 are the tightest columns, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 11 rows. Row 7 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.
The whole middle game reduces to two sweeps. Sweep the twelve colors looking for one with a single square left; sweep the twelve rows and twelve columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 12x12 solve from a stalled one.
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 #46 is rated easy
Ratings track how generous the opening is. On #46 you get six regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 35-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.
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.
A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.
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 #46, start hunting for it near row 3.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 9 queens in the middle, including 1 in a corner. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #46?
Puzzle #46 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and six of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #46 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #46, 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 #46?
Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.