9x9 Queens Puzzle #211

About 9x9 Queens puzzle #211

9x9 puzzle #211 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.

Puzzle #211 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 #211 are: pale yellow (2 squares), khaki (4 squares), orange (7 squares), blue (10 squares), mint (10 squares), slate (10 squares), indigo (11 squares), red (12 squares), amber (15 squares). An even split would hand every color 9 squares, so the pale yellow region is running 7 squares under that average while the amber region is the roomiest area on the grid with 15 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Pale yellow is working with 2 squares while amber has 15 to spare, a gap of 13. Attack the cramped end and the roomy regions tend to resolve themselves.

Shape matters as much as size. The pale yellow region is the most compact color here, filling 100% of the 1×2 box that encloses it, while orange is the most scattered at 47% of its 3×5 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

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.

Pale yellow covers 2 squares, spanning 1 row and 2 columns (rows 7–7, columns 6–7). It never leaves row 7, so its queen is locked to that row.

Khaki occupies 4 squares within rows 7–8, columns 8–9.

Orange: 7 squares across rows 7–9, columns 5–9.

Blue covers 10 squares, spanning 4 rows and 3 columns (rows 1–4, columns 3–5).

Mint occupies 10 squares within rows 3–6, columns 7–9.

Slate: 10 squares across rows 5–8, columns 2–6.

Indigo covers 11 squares, spanning 5 rows and 3 columns (rows 1–5, columns 1–3).

Red occupies 12 squares within rows 6–9, columns 1–6.

Amber: 15 squares across rows 1–5, columns 5–9.

Where to start on this board

Pale yellow is the natural opening on #211: it is the smallest region on the grid at 2 squares, boxed into rows 7–7 and columns 6–7. Since one of those squares must hold a queen, that block of 1 row and 2 columns is the most contested part of the board.

Because pale yellow never leaves row 7, its queen must be in that row — and that instantly settles row 7 for everyone else. Sweep along it and X out every square that is not pale yellow. 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 is no more than two rows tall, so its queen is 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 #211

A quick way to find the next move is to count how many different colors touch each line. On this board, row 9 is the narrowest, touched by only 2 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 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 8 rows. By contrast row 7 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 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.

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 #211 is rated medium

The medium label comes out of four measurable things about #211: one of its regions is locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 2 colors, and its largest and smallest regions differ by 13. 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.

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 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 row 9 on this board.

As a sanity check rather than a spoiler: the finished solution to #211 puts 4 queens on the outer border and 5 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.