12x12 Queens Puzzle #3

About 12x12 Queens puzzle #3

12x12 puzzle #3 is rated hard. Twelve colored regions cover its 144 squares, and your job is to seat twelve queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

Like every board on this site, #3 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.

How the colors divide this board

From smallest to largest, the regions on puzzle #3 are: blue (1 square), rose (1 square), teal (3 squares), slate (6 squares), amber (7 squares), indigo (10 squares), red (12 squares), orange (12 squares), khaki (14 squares), sea green (15 squares), mint (24 squares), pale yellow (39 squares). An even split would hand every color 12 squares, so the blue region is running 11 squares under that average while the pale yellow region is the roomiest area on the grid with 39 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Blue has only 1 place to put its queen; pale yellow has 39. The spread between them on this board is 38 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 blue is the most solid, filling 100% of its 1×1 box, and mint the most diffuse at 44% of its 6×9 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 hard board.

Blue: 1 square across rows 3–3, columns 12–12. It never leaves row 3, so its queen is locked to that row.

Rose covers 1 square, spanning 1 row and 1 column (rows 12–12, columns 6–6). It never leaves row 12, so its queen is locked to that row.

Teal occupies 3 squares within rows 11–12, columns 1–2.

Slate: 6 squares across rows 4–5, columns 9–12.

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

Indigo occupies 10 squares within rows 1–4, columns 10–12.

Red: 12 squares across rows 4–6, columns 2–6.

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

Khaki occupies 14 squares within rows 6–10, columns 1–4.

Sea green: 15 squares across rows 6–12, columns 2–5.

Mint covers 24 squares, spanning 6 rows and 9 columns (rows 1–6, columns 1–9).

Pale yellow occupies 39 squares within rows 4–12, columns 6–12.

Where to start on this board

Begin where the choices are fewest. On this board that means blue, which owns just 1 square inside a 1×1 window running from row 3 to row 3 and column 12 to column 12. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

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. Rose, amber, slate and teal 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 #3

A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Columns 1, 7 and 8 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 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 11 rows. By contrast row 12 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.

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.

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 #3 is rated hard

The hard label comes out of four measurable things about #3: four of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 38. 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.

If #3 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.

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 #3 the place to look for one is around row 1.

As a sanity check rather than a spoiler: the finished solution to #3 puts 4 queens on the outer border and 8 queens in the interior, with none in a 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.