12x12 Queens Puzzle #265

About 12x12 Queens puzzle #265

Puzzle #265 is one of the medium-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.

There is exactly one way to finish #265, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these twelve regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #265 are: blue (1 square), orange (1 square), pale yellow (2 squares), indigo (5 squares), teal (5 squares), rose (8 squares), slate (9 squares), khaki (18 squares), amber (19 squares), sea green (21 squares), mint (22 squares), red (33 squares). An even split would hand every color 12 squares, so the blue region is running 11 squares under that average while the red 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 blue 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. Blue is the densest region on this grid at 100% of a 1×1 frame; mint is the loosest, occupying only 46% of its 6×8 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.

Blue covers 1 square, spanning 1 row and 1 column (rows 3–3, columns 3–3). It never leaves row 3, so its queen is locked to that row.

Orange occupies 1 square within rows 9–9, columns 11–11. It never leaves row 9, so its queen is locked to that row.

Pale yellow: 2 squares across rows 6–7, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Indigo covers 5 squares, spanning 3 rows and 3 columns (rows 1–3, columns 6–8).

Teal occupies 5 squares within rows 9–11, columns 10–12.

Rose: 8 squares across rows 10–12, columns 9–12.

Slate covers 9 squares, spanning 5 rows and 2 columns (rows 1–5, columns 1–2).

Khaki occupies 18 squares within rows 5–9, columns 8–12.

Amber: 19 squares across rows 1–4, columns 7–12.

Sea green covers 21 squares, spanning 6 rows and 5 columns (rows 7–12, columns 4–8).

Mint occupies 22 squares within rows 1–6, columns 2–9.

Red: 33 squares across rows 5–12, columns 1–7.

Where to start on this board

Open on the tightest color. Here that is blue, confined to 1 square between rows 3 and 3 and columns 3 and 3. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #265.

Because blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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. Orange and pale yellow 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 #265

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 4, 8 and 12 are the narrowest, touched by only 3 colors. Columns 1, 2, 3, 4 and 5 are the tightest columns, with 3 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 5 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.

Two checks, run alternately, will carry you through the 144 squares of this grid. First: has any of the twelve regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 12x12 boards almost always run one of those checks and forget the other.

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 #265 is rated medium

The medium label comes out of four measurable things about #265: five 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 32. 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.

Stuck on #265 despite the medium rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.

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 #265 is row 4.

As a sanity check rather than a spoiler: the finished solution to #265 puts 3 queens on the outer border and 9 queens in the interior, 1 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.