12x12 Queens Puzzle #215

About 12x12 Queens puzzle #215

12x12 puzzle #215 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, #215 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 #215 are: indigo (1 square), slate (1 square), rose (4 squares), orange (5 squares), mint (6 squares), amber (7 squares), red (12 squares), teal (12 squares), sea green (19 squares), khaki (20 squares), pale yellow (23 squares), blue (34 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 34 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 34. The spread between them on this board is 33 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×1 box that encloses it, while khaki is the most scattered at 48% of its 7×6 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.

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.

Slate: 1 square across rows 5–5, columns 4–4. It never leaves row 5, so its queen is locked to that row.

Rose covers 4 squares, spanning 3 rows and 2 columns (rows 10–12, columns 6–7).

Orange occupies 5 squares within rows 9–12, columns 11–12.

Mint: 6 squares across rows 2–5, columns 1–2.

Amber covers 7 squares, spanning 2 rows and 5 columns (rows 1–2, columns 7–11).

Red occupies 12 squares within rows 4–8, columns 5–9.

Teal: 12 squares across rows 10–12, columns 1–6.

Sea green covers 19 squares, spanning 5 rows and 5 columns (rows 8–12, columns 7–11).

Khaki occupies 20 squares within rows 5–11, columns 1–6.

Pale yellow: 23 squares across rows 2–9, columns 9–12.

Blue covers 34 squares, spanning 8 rows and 8 columns (rows 1–8, columns 1–8).

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 12 to column 12. 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. Slate and amber 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 #215

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 3 are the narrowest, touched by only 3 colors. Columns 3, 10 and 12 are the tightest columns, with 3 colors.

A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 11 rows. Row 5 is the crowded extreme at 6 colors — worth revisiting later, not now.

Working through the middle game

After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.

Keep alternating between two views of the board. In the region view you ask which of the twelve colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 12x12 board unwinds.

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 #215 is rated easy

The easy label comes out of four measurable things about #215: four 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 33. 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.

If #215 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 #215 the place to look for one is around row 1.

A non-spoiling distribution check: solved, #215 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.