10x10 Queens Puzzle #212

About 10x10 Queens puzzle #212

This is puzzle number 212 in the 10x10 collection, rated easy. The grid holds 100 squares split between ten colored regions, and a finished board carries ten queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

Puzzle #212 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 #212 are: slate (1 square), sea green (2 squares), blue (4 squares), indigo (5 squares), orange (6 squares), pale yellow (10 squares), amber (11 squares), khaki (11 squares), red (18 squares), mint (32 squares). An even split would hand every color 10 squares, so the slate region is running 9 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 slate 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.

Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×5 frame; pale yellow is the loosest, occupying only 50% of its 4×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

Region by region

Here is the full footprint of every color on puzzle #212, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Slate occupies 1 square within rows 5–5, columns 7–7. It never leaves row 5, so its queen is locked to that row.

Sea green: 2 squares across rows 9–10, columns 5–5. It sits wholly inside column 5, so its queen must be in that column.

Blue covers 4 squares, spanning 3 rows and 2 columns (rows 3–5, columns 8–9).

Indigo occupies 5 squares within rows 1–1, columns 2–6. It never leaves row 1, so its queen is locked to that row.

Orange: 6 squares across rows 9–10, columns 1–4.

Pale yellow covers 10 squares, spanning 4 rows and 5 columns (rows 5–8, columns 2–6).

Amber occupies 11 squares within rows 1–3, columns 5–10.

Khaki: 11 squares across rows 7–10, columns 5–9.

Red covers 18 squares, spanning 7 rows and 5 columns (rows 4–10, columns 6–10).

Mint occupies 32 squares within rows 1–9, columns 1–7.

Where to start on this board

Slate is the natural opening on #212: it is the smallest region on the grid at 1 square, boxed into rows 5–5 and columns 7–7. 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 slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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, orange and sea green 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 #212

A quick way to find the next move is to count how many different colors touch each line. On this board, row 2 is the narrowest, touched by only 2 colors. Columns 1 and 10 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 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 9 rows. By contrast row 9 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.

Two checks, run alternately, will carry you through the 100 squares of this grid. First: has any of the ten regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 10x10 boards almost always run one of those checks and forget the other.

Resist the plausible-looking queen. On #212 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 #212 is rated easy

The rating reflects how much the board concedes up front. Puzzle #212 has four regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 31 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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 10x10

At 10x10 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.

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 #212 is row 2.

As a sanity check rather than a spoiler: the finished solution to #212 puts 4 queens on the outer border and 6 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.