12x12 Queens Puzzle #55
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 #55
What follows is a complete solve of puzzle #55, 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 line confined to one region.
Techniques this board needs: last square in a region, region shadow, region confined to one line, line confined to one region, last square in a row.
Show the full walkthrough (9 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: indigo has one square left, so its queen takes row 1, column 2; amber has one square left, so its queen takes row 2, column 5; pale yellow has one square left, so its queen takes row 7, column 1; khaki has one square left, so its queen takes row 8, column 12. 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 12.
Step 2. Red is down to 3 candidates in rows 5 and 6, columns 7 and 8, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about red itself.
Step 3. All of the rose region's remaining squares (3 open squares) lie in row 12. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.
The board after step 3 — 4 of 12 queens placed. Step 4. Wherever orange 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 5. Every open square left in row 3 (6 open squares) belongs to the blue region. Since that row must contain a queen, blue is the colour that supplies it — so blue squares anywhere else on the board are out, 8 squares in all.
Step 6. Look along row 4: one square survives, row 4, column 11, which belongs to mint. A row cannot be left empty, so the queen goes there. Everything else in row 4, in column 11, in mint and in the ring around that square is now dead.
The board after step 6 — 5 of 12 queens placed. Step 7. All of the teal region's remaining squares (2 open squares) lie in column 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 3 squares come off.
Step 8. Orange is down to 2 candidates in rows 9 and 11, columns 8 and 9, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about orange itself.
Step 9. 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: rose has one square left, so its queen takes row 12, column 8; red has one square left, so its queen takes row 6, column 7; orange has one square left, so its queen takes row 9, column 9; blue has one square left, so its queen takes row 3, column 3; teal has one square left, so its queen takes row 11, column 10; sea green has one square left, so its queen takes row 10, column 6; slate has one square left, so its queen takes row 5, 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 5, column 4.
The board after step 9 — 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 #55
12x12 puzzle #55 is rated easy. Twelve colored regions cover its 144 squares, and your job is to seat twelve queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Like every board on this site, #55 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.
How the colors divide this board
From smallest to largest, the regions on puzzle #55 are: indigo (1 square), red (3 squares), khaki (3 squares), rose (3 squares), pale yellow (4 squares), amber (5 squares), teal (9 squares), orange (15 squares), mint (16 squares), slate (22 squares), sea green (25 squares), blue (38 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the blue region is the roomiest area on the grid with 38 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 1 place to put its queen; blue has 38. The spread between them on this board is 37 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and orange the most diffuse at 60% of its 5×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #55 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 2–2). It never leaves row 1, so its queen is locked to that row.
Red occupies 3 squares within rows 5–6, columns 7–8.
Khaki: 3 squares across rows 7–8, columns 11–12.
Rose covers 3 squares, spanning 1 row and 3 columns (rows 12–12, columns 8–10). It never leaves row 12, so its queen is locked to that row.
Pale yellow occupies 4 squares within rows 7–9, columns 1–2.
Amber: 5 squares across rows 1–2, columns 4–7.
Teal covers 9 squares, spanning 4 rows and 3 columns (rows 9–12, columns 10–12).
Orange occupies 15 squares within rows 7–11, columns 7–11.
Mint: 16 squares across rows 1–6, columns 9–12.
Slate covers 22 squares, spanning 6 rows and 6 columns (rows 5–10, columns 1–6).
Sea green occupies 25 squares within rows 8–12, columns 1–8.
Blue: 38 squares across rows 1–5, columns 1–11.
Where to start on this board
Begin where the choices are fewest. On this board that means indigo, which owns just 1 square inside a 1×1 window running from row 1 to row 1 and column 2 to column 2. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
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. Rose, amber, red and khaki 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 #55
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 4 are the narrowest, touched by only 2 colors. Columns 3 and 12 are the tightest columns, with 3 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 9 is the crowded extreme at 5 colors — worth revisiting later, not now.
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.
With twelve queens to seat and twelve regions competing over 144 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 12x12 boards quickly and players who stall halfway.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #55 is rated easy
What earns a board its rating is the shape of its opening. Here that opening offers three single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 37. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
If #55 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.
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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #55 the place to look for one is around row 3.
A non-spoiling distribution check: solved, #55 carries 4 queens on the perimeter against 8 queens inside, with the corners empty. 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 #55?
Puzzle #55 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #55 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #55, 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 #55?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.