12x12 Queens Puzzle #240

About 12x12 Queens puzzle #240

This is puzzle number 240 in the 12x12 collection, rated hard. The grid holds 144 squares split between twelve colored regions, and a finished board carries twelve queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

Puzzle #240 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.

How the colors divide this board

From smallest to largest, the regions on puzzle #240 are: indigo (1 square), khaki (1 square), teal (3 squares), pale yellow (6 squares), rose (9 squares), red (10 squares), amber (13 squares), slate (14 squares), blue (18 squares), mint (18 squares), sea green (18 squares), orange (33 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the orange 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 indigo and orange 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 #240, packing 100% of its 1×1 bounding box, whereas pale yellow straggles across its 4×3 box at just 50%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Below is every color on #240 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: 1 square across rows 1–1, columns 12–12. It never leaves row 1, so its queen is locked to that row.

Khaki covers 1 square, spanning 1 row and 1 column (rows 8–8, columns 6–6). It never leaves row 8, so its queen is locked to that row.

Teal occupies 3 squares within rows 10–12, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Pale yellow: 6 squares across rows 4–7, columns 8–10.

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

Red occupies 10 squares within rows 5–7, columns 5–8.

Amber: 13 squares across rows 1–3, columns 6–11.

Slate covers 14 squares, spanning 8 rows and 3 columns (rows 2–9, columns 10–12).

Blue occupies 18 squares within rows 1–6, columns 1–4.

Mint: 18 squares across rows 1–6, columns 3–8.

Sea green covers 18 squares, spanning 6 rows and 5 columns (rows 7–12, columns 7–11).

Orange occupies 33 squares within rows 7–12, columns 1–7.

Where to start on this board

The most constrained color is where a solve should begin, and on #240 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 12 to column 12 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.

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. Khaki 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 #240

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3, 4, 8, 9, 10, 11 and 12 are the narrowest, touched by only 4 colors. Column 2 is the tightest column, with 2 colors.

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 4 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 7, meeting 5 colors, is the opposite case — the busiest line on #240 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.

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.

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 #240 is rated hard

What earns a board its rating is the shape of its opening. Here that opening offers five single-line regions, a 1-square smallest color, a tightest row touched by 4 colors, and a region-size spread of 32. 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.

When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.

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 #240 is column 2.

Here is a shape check that gives nothing away: when #240 is solved, 3 queens sit on the outer edge of the grid and 9 sit inside it, with 1 queen in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.