12x12 Queens Puzzle #40

About 12x12 Queens puzzle #40

This is puzzle number 40 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.

There is exactly one way to finish #40, 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 #40 are: sea green (1 square), rose (1 square), indigo (2 squares), teal (7 squares), mint (8 squares), blue (9 squares), amber (10 squares), slate (12 squares), red (15 squares), khaki (24 squares), pale yellow (27 squares), orange (28 squares). An even split would hand every color 12 squares, so the sea green region is running 11 squares under that average while the orange 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 sea green 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.

Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×2 box that encloses it, while pale yellow is the most scattered at 56% of its 8×6 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.

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.

Sea green covers 1 square, spanning 1 row and 1 column (rows 10–10, columns 4–4). It never leaves row 10, so its queen is locked to that row.

Rose occupies 1 square within rows 12–12, columns 2–2. It never leaves row 12, so its queen is locked to that row.

Indigo: 2 squares across rows 1–1, columns 10–11. It never leaves row 1, so its queen is locked to that row.

Teal covers 7 squares, spanning 3 rows and 4 columns (rows 10–12, columns 9–12).

Mint occupies 8 squares within rows 1–4, columns 8–10.

Blue: 9 squares across rows 1–5, columns 10–12.

Amber covers 10 squares, spanning 3 rows and 5 columns (rows 1–3, columns 3–7).

Slate occupies 12 squares within rows 2–5, columns 4–8.

Red: 15 squares across rows 1–6, columns 1–3.

Khaki covers 24 squares, spanning 8 rows and 5 columns (rows 5–12, columns 1–5).

Pale yellow occupies 27 squares within rows 4–11, columns 7–12.

Orange: 28 squares across rows 5–12, columns 4–9.

Where to start on this board

The most constrained color is where a solve should begin, and on #40 that is sea green with 1 square. It reaches from row 10 to row 10 and from column 4 to column 4 — 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 sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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. Indigo and rose 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 #40

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 7, 8 and 9 are the narrowest, touched by only 3 colors. Column 1 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 5 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.

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 #40 is rated hard

The hard label comes out of four measurable things about #40: 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 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.

Stuck on #40 despite the hard 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 #40 is column 1.

As a sanity check rather than a spoiler: the finished solution to #40 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.