12x12 Queens Puzzle #181

About 12x12 Queens puzzle #181

Puzzle #181 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, #181 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 #181 are: indigo (2 squares), sea green (2 squares), mint (5 squares), slate (6 squares), teal (7 squares), pale yellow (8 squares), orange (11 squares), rose (11 squares), blue (13 squares), red (13 squares), khaki (24 squares), amber (42 squares). An even split would hand every color 12 squares, so the indigo region is running 10 squares under that average while the amber region is the roomiest area on the grid with 42 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 indigo, with 2 squares, is under far more pressure than amber with 42. A 40-square gap like this one rewards players who start where the choices are fewest.

Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #181, packing 100% of its 1×2 bounding box, whereas amber straggles across its 9×10 box at just 47%. 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.

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.

Sea green occupies 2 squares within rows 10–10, columns 9–10. It never leaves row 10, so its queen is locked to that row.

Mint: 5 squares across rows 4–6, columns 3–5.

Slate covers 6 squares, spanning 3 rows and 3 columns (rows 5–7, columns 1–3).

Teal occupies 7 squares within rows 9–12, columns 8–10.

Pale yellow: 8 squares across rows 7–10, columns 1–3.

Orange covers 11 squares, spanning 4 rows and 4 columns (rows 9–12, columns 9–12).

Rose occupies 11 squares within rows 10–12, columns 1–5.

Blue: 13 squares across rows 1–5, columns 1–4.

Red covers 13 squares, spanning 4 rows and 5 columns (rows 5–8, columns 8–12).

Khaki occupies 24 squares within rows 6–12, columns 4–10.

Amber: 42 squares across rows 1–9, columns 3–12.

Where to start on this board

Open on the tightest color. Here that is indigo, confined to 2 squares between rows 1 and 1 and columns 11 and 12. Its queen has nowhere else to go, which means those 1 row and 2 columns carry more constraint than anywhere else on #181.

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. Sea green is no more than two rows tall, so its queen is 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 #181

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 6 and 7 are the tightest columns, with 2 colors.

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 10, meeting 6 colors, is the opposite case — the busiest line on #181 and the last one you should expect to crack.

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.

A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.

Why puzzle #181 is rated easy

The easy label comes out of four measurable things about #181: two of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 2 colors, and its largest and smallest regions differ by 40. 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.

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 #181, start hunting for it near row 2.

Here is a shape check that gives nothing away: when #181 is solved, 4 queens sit on the outer edge of the grid and 8 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.