12x12 Queens Puzzle #298

About 12x12 Queens puzzle #298

Board #298 of the 12x12 bank carries a easy 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.

Like every board on this site, #298 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 #298 are: blue (2 squares), mint (2 squares), rose (3 squares), khaki (7 squares), slate (11 squares), orange (11 squares), sea green (11 squares), teal (12 squares), pale yellow (16 squares), indigo (19 squares), red (23 squares), amber (27 squares). An even split would hand every color 12 squares, so the blue region is running 10 squares under that average while the amber region is the roomiest area on the grid with 27 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 2 places to put its queen; amber has 27. The spread between them on this board is 25 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Look at arrangement as well as area. Blue is the densest region on this grid at 100% of a 2×1 frame; teal is the loosest, occupying only 48% of its 5×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

Region by region

Below is every color on #298 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.

Blue covers 2 squares, spanning 2 rows and 1 column (rows 3–4, columns 6–6). It sits wholly inside column 6, so its queen must be in that column.

Mint occupies 2 squares within rows 4–4, columns 7–8. It never leaves row 4, so its queen is locked to that row.

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

Khaki covers 7 squares, spanning 3 rows and 3 columns (rows 8–10, columns 1–3).

Slate occupies 11 squares within rows 4–7, columns 1–5.

Orange: 11 squares across rows 9–12, columns 9–12.

Sea green covers 11 squares, spanning 3 rows and 6 columns (rows 10–12, columns 1–6).

Teal occupies 12 squares within rows 8–12, columns 7–11.

Pale yellow: 16 squares across rows 1–9, columns 10–12.

Indigo covers 19 squares, spanning 4 rows and 6 columns (rows 1–4, columns 1–6).

Red occupies 23 squares within rows 5–11, columns 2–7.

Amber: 27 squares across rows 1–8, columns 7–11.

Where to start on this board

Blue is the natural opening on #298: it is the smallest region on the grid at 2 squares, boxed into rows 3–4 and columns 6–6. Since one of those squares must hold a queen, that block of 2 rows and 1 column is the most contested part of the board.

Because blue stays inside column 6, its queen belongs to that column and no other region may use it. Cross out the rest of column 6 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.

Work outward to the other flat regions next. Mint 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 #298

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

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 4, meeting 6 colors, is the opposite case — the busiest line on #298 and the last one you should expect to crack.

Working through the middle game

The middle of #298 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.

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 #298 is rated easy

What earns a board its rating is the shape of its opening. Here that opening offers three single-line regions, a 2-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 25. 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.

If #298 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 #298 the place to look for one is around column 12.

Here is a shape check that gives nothing away: when #298 is solved, 4 queens sit on the outer edge of the grid and 8 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.