12x12 Queens Puzzle #57
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 24-step reasoning is written out below.
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Step-by-step solution for puzzle #57
What follows is a complete solve of puzzle #57, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 12 queen placements and 12 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, line confined to one region, last square in a row, pairing argument on columns.
Show the full walkthrough (19 steps) — contains the solution
Step 1. Only row 6, column 12 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 12, the rest of the red region and every square adjacent to it — diagonals included.
Step 2. The indigo region is down to one open square, at row 1, column 11. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 11, the remaining indigo squares, and the ring of neighbours around row 1, column 11.
Step 3. Only row 2, column 7 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 7, in amber and in the ring around that square is now dead.
Step 4. Wherever khaki ends up among its 3 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
Step 5. Teal is down to 5 candidates in rows 11 and 12, columns 1–3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about teal itself.
The board after step 5 — 3 of 12 queens placed. Step 6. The rose region has 4 squares left, spread across rows 11 and 12, columns 4–6. 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. Teal and rose are now boxed into rows 11 and 12 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 6 squares at once.
Step 8. The orange region is down to one open square, at row 9, column 10. Every colour holds exactly one queen, so that square is forced. Row 9 and column 10 are now spoken for, along with the squares touching that queen.
Step 9. Only row 8, column 5 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 5, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 10. Wherever blue 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.
The board after step 10 — 5 of 12 queens placed. Step 11. Rose is down to 2 candidates in rows 11 and 12, columns 4 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about rose itself.
Step 12. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 6. Follow the forced moves that would create and the pale yellow region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 13. Wherever blue ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 14. Every open square left in column 9 (2 open squares) belongs to the blue region. Since that column must contain a queen, blue is the colour that supplies it — so blue squares anywhere else on the board are out, 1 square in all.
The board after step 14 — 5 of 12 queens placed. Step 15. Row 4 has only mint squares left (4 open squares). The row needs a queen and only mint can provide one, which pins that colour to this row and kills its squares elsewhere: 7 squares.
Step 16. Every open square left in row 5 (2 open squares) belongs to the slate region. Since that row must contain a queen, slate is the colour that supplies it — so slate squares anywhere else on the board are out, 3 squares in all.
Step 17. Look along row 3: one square survives, row 3, column 9, which belongs to blue. A row cannot be left empty, so the queen goes there. Everything else in row 3, in column 9, in blue and in the ring around that square is now dead.
Step 18. Here is the pairing argument. The slate and pale yellow regions have no open squares outside columns 6 and 8 — 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: 1 square come off in one stroke.
Step 19. 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 4; teal has one square left, so its queen takes row 11, column 1; sea green has one square left, so its queen takes row 10, column 3; mint has one square left, so its queen takes row 4, column 2; pale yellow has one square left, so its queen takes row 7, column 8; slate has one square left, so its queen takes row 5, column 6. 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 5, column 6.
The board after step 19 — 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 #57
Puzzle #57 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.
As with every puzzle in this collection, #57 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 #57 are: red (1 square), indigo (2 squares), khaki (3 squares), amber (5 squares), teal (5 squares), rose (5 squares), sea green (15 squares), orange (17 squares), pale yellow (18 squares), mint (19 squares), slate (21 squares), blue (33 squares). An even split would hand every color 12 squares, so the red region is running 11 squares under that average while the blue region is the roomiest area on the grid with 33 squares.
Every color takes one queen and one only, whatever its size — which is why the 32 square difference between red and blue is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #57, packing 100% of its 1×2 bounding box, whereas pale yellow straggles across its 7×6 box at just 43%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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.
Red: 1 square across rows 6–6, columns 12–12. It never leaves row 6, so its queen is locked to that row.
Indigo covers 2 squares, spanning 1 row and 2 columns (rows 1–1, columns 11–12). It never leaves row 1, so its queen is locked to that row.
Khaki occupies 3 squares within rows 8–9, columns 4–5.
Amber: 5 squares across rows 1–2, columns 5–8.
Teal covers 5 squares, spanning 2 rows and 3 columns (rows 11–12, columns 1–3).
Rose occupies 5 squares within rows 11–12, columns 4–7.
Sea green: 15 squares across rows 6–11, columns 1–4.
Orange covers 17 squares, spanning 4 rows and 7 columns (rows 9–12, columns 6–12).
Pale yellow occupies 18 squares within rows 4–10, columns 5–10.
Mint: 19 squares across rows 1–5, columns 1–4.
Slate covers 21 squares, spanning 7 rows and 7 columns (rows 2–8, columns 2–8).
Blue occupies 33 squares within rows 1–9, columns 6–12.
Where to start on this board
Open on the tightest color. Here that is red, confined to 1 square between rows 6 and 6 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 #57.
Because red never leaves row 6, its queen must be in that row — and that instantly settles row 6 for everyone else. Sweep along it and X out every square that is not red. 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. Indigo, amber, khaki 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 #57
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3, 10 and 12 are the narrowest, touched by only 3 colors. Columns 1, 9, 10 and 11 are the tightest columns, with 3 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 11 rows. By contrast row 9 meets 5 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
Working through the middle game
The middle of #57 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.
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 #57 is rated easy
The easy label comes out of four measurable things about #57: three of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 32. 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.
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.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #57 is row 3.
As a sanity check rather than a spoiler: the finished solution to #57 puts 4 queens on the outer border and 8 queens in the interior, with none in a 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 #57?
Puzzle #57 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #57 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #57, 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 #57?
Start with the red region — the most constrained color on this board, with 1 square spanning rows 6–6 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.