12x12 Queens Puzzle #152

About 12x12 Queens puzzle #152

This is puzzle number 152 in the 12x12 collection, rated easy. 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 #152 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 #152 are: indigo (1 square), sea green (2 squares), red (4 squares), teal (7 squares), amber (9 squares), pale yellow (11 squares), rose (11 squares), khaki (14 squares), orange (16 squares), blue (17 squares), slate (20 squares), mint (32 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the mint region is the roomiest area on the grid with 32 squares.

Every color takes one queen and one only, whatever its size — which is why the 31 square difference between indigo and mint 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 1×1 box that encloses it, while khaki is the most scattered at 56% of its 5×5 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

Below is every color on #152 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

Indigo occupies 1 square within rows 1–1, columns 5–5. It never leaves row 1, so its queen is locked to that row.

Sea green: 2 squares across rows 10–10, columns 8–9. It never leaves row 10, so its queen is locked to that row.

Red covers 4 squares, spanning 3 rows and 2 columns (rows 5–7, columns 1–2).

Teal occupies 7 squares within rows 10–12, columns 1–4.

Amber: 9 squares across rows 1–4, columns 6–9.

Pale yellow covers 11 squares, spanning 4 rows and 3 columns (rows 7–10, columns 6–8).

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

Khaki: 14 squares across rows 7–11, columns 1–5.

Orange covers 16 squares, spanning 7 rows and 4 columns (rows 6–12, columns 9–12).

Blue occupies 17 squares within rows 1–5, columns 1–5.

Slate: 20 squares across rows 3–9, columns 3–7.

Mint covers 32 squares, spanning 9 rows and 5 columns (rows 1–9, columns 8–12).

Where to start on this board

The most constrained color is where a solve should begin, and on #152 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 5 to column 5 — 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. Sea green 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 #152

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2 and 12 are the narrowest, touched by only 3 colors. Columns 11 and 12 are the tightest columns, 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 7 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

After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.

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 #152 is rated easy

What earns a board its rating is the shape of its opening. Here that opening offers three single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 31. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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 #152 is column 11.

As a sanity check rather than a spoiler: the finished solution to #152 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.