12x12 Queens Puzzle #280

About 12x12 Queens puzzle #280

This is puzzle number 280 in the 12x12 collection, rated medium. 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 #280 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 #280 are: indigo (1 square), pale yellow (1 square), amber (5 squares), orange (5 squares), rose (5 squares), teal (9 squares), khaki (11 squares), mint (15 squares), slate (17 squares), blue (20 squares), sea green (27 squares), red (28 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the red region is the roomiest area on the grid with 28 squares.

Every color takes one queen and one only, whatever its size — which is why the 27 square difference between indigo and red 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.

Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; khaki is the loosest, occupying only 44% of its 5×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

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 medium board.

Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 12–12). It never leaves row 1, so its queen is locked to that row.

Pale yellow occupies 1 square within rows 7–7, columns 2–2. It never leaves row 7, so its queen is locked to that row.

Amber: 5 squares across rows 1–2, columns 10–12.

Orange covers 5 squares, spanning 3 rows and 2 columns (rows 7–9, columns 7–8).

Rose occupies 5 squares within rows 9–12, columns 1–2.

Teal: 9 squares across rows 9–12, columns 3–5.

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

Mint occupies 15 squares within rows 1–6, columns 1–3.

Slate: 17 squares across rows 3–8, columns 10–12.

Blue covers 20 squares, spanning 6 rows and 4 columns (rows 1–6, columns 6–9).

Sea green occupies 27 squares within rows 6–12, columns 6–12.

Red: 28 squares across rows 1–11, columns 3–7.

Where to start on this board

The most constrained color is where a solve should begin, and on #280 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. Pale yellow 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 #280

A quick way to find the next move is to count how many different colors touch each line. On this board, row 12 is the narrowest, touched by only 3 colors. Column 9 is the tightest column, with 2 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 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

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.

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 #280 is rated medium

The medium label comes out of four measurable things about #280: 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 27. Boards that score tight on those measures give up their first queen fast.

Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.

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 #280 is column 9.

As a sanity check rather than a spoiler: the finished solution to #280 puts 2 queens on the outer border and 10 queens in the interior, 2 of them in an actual 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.