12x12 Queens Puzzle #6

About 12x12 Queens puzzle #6

Board #6 of the 12x12 bank carries a medium rating. Its 144 squares are shared between twelve colored regions, and the solution places twelve queens: one per row, one per column, one per region, none of them side by side or corner to corner.

As with every puzzle in this collection, #6 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 #6 are: indigo (1 square), mint (2 squares), rose (2 squares), khaki (7 squares), amber (8 squares), red (9 squares), pale yellow (9 squares), orange (9 squares), slate (12 squares), teal (13 squares), blue (30 squares), sea green (42 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the sea green region is the roomiest area on the grid with 42 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; sea green has 42. The spread between them on this board is 41 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and orange the most diffuse at 45% of its 4×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

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: 1 square across rows 1–1, columns 12–12. It never leaves row 1, so its queen is locked to that row.

Mint covers 2 squares, spanning 2 rows and 1 column (rows 4–5, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Rose occupies 2 squares within rows 12–12, columns 7–8. It never leaves row 12, so its queen is locked to that row.

Khaki: 7 squares across rows 6–8, columns 8–10.

Amber covers 8 squares, spanning 4 rows and 3 columns (rows 1–4, columns 3–5).

Red occupies 9 squares within rows 3–6, columns 10–12.

Pale yellow: 9 squares across rows 6–12, columns 1–2.

Orange covers 9 squares, spanning 4 rows and 5 columns (rows 6–9, columns 8–12).

Slate occupies 12 squares within rows 1–7, columns 1–3.

Teal: 13 squares across rows 9–12, columns 8–12.

Blue covers 30 squares, spanning 5 rows and 9 columns (rows 1–5, columns 4–12).

Sea green occupies 42 squares within rows 5–12, columns 2–9.

Where to start on this board

Indigo is the natural opening on #6: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 12–12. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of 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. Rose and mint 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 #6

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 10 and 11 are the narrowest, touched by only 3 colors. Column 6 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 6 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.

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.

The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.

Why puzzle #6 is rated medium

The medium label comes out of four measurable things about #6: 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 41. 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.

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.

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 #6 the place to look for one is around column 6.

As a sanity check rather than a spoiler: the finished solution to #6 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.