9x9 Queens Puzzle #259

About 9x9 Queens puzzle #259

9x9 puzzle #259 is rated medium. Nine colored regions cover its 81 squares, and your job is to seat nine queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

Like every board on this site, #259 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 #259 are: blue (1 square), khaki (1 square), indigo (3 squares), orange (5 squares), mint (11 squares), slate (12 squares), red (14 squares), amber (16 squares), pale yellow (18 squares). An even split would hand every color 9 squares, so the blue region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 18 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Blue is working with 1 square while pale yellow has 18 to spare, a gap of 17. Attack the cramped end and the roomy regions tend to resolve themselves.

Shape matters as much as size. The blue region is the most compact color here, filling 100% of the 1×1 box that encloses it, while mint is the most scattered at 55% of its 5×4 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 medium board.

Blue covers 1 square, spanning 1 row and 1 column (rows 3–3, columns 9–9). It never leaves row 3, so its queen is locked to that row.

Khaki occupies 1 square within rows 8–8, columns 2–2. It never leaves row 8, so its queen is locked to that row.

Indigo: 3 squares across rows 1–2, columns 8–9.

Orange covers 5 squares, spanning 2 rows and 4 columns (rows 8–9, columns 1–4).

Mint occupies 11 squares within rows 1–5, columns 6–9.

Slate: 12 squares across rows 4–8, columns 1–4.

Red covers 14 squares, spanning 6 rows and 3 columns (rows 4–9, columns 7–9).

Amber occupies 16 squares within rows 1–3, columns 1–6.

Pale yellow: 18 squares across rows 4–9, columns 3–7.

Where to start on this board

Blue is the natural opening on #259: it is the smallest region on the grid at 1 square, boxed into rows 3–3 and columns 9–9. 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 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. Khaki, indigo and orange 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 #259

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3, 6, 7 and 9 are the narrowest, touched by only 3 colors. Column 5 is the tightest column, with 2 colors.

A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 8 rows. Row 8 is the crowded extreme at 5 colors — worth revisiting later, not now.

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.

With nine queens to seat and nine regions competing over 81 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 9x9 boards quickly and players who stall halfway.

Resist the plausible-looking queen. On #259 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.

Why puzzle #259 is rated medium

The medium label comes out of four measurable things about #259: 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 17. Boards that score tight on those measures give up their first queen fast.

Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.

If #259 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 9x9

At 9x9 a full re-scan after every move starts to get slow, so work locally instead. Once a queen goes down, check only the regions, rows and columns that placement actually touched — that is where new information appeared. Fall back to a complete sweep only when the local pass turns up nothing, and your solve times will drop noticeably.

Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around column 5 on this board.

A non-spoiling distribution check: solved, #259 carries 4 queens on the perimeter against 5 queens inside, with the corners empty. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.