10x10 Queens Puzzle #96

About 10x10 Queens puzzle #96

This is puzzle number 96 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 #96 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 #96 are: indigo (1 square), red (1 square), khaki (1 square), blue (4 squares), amber (7 squares), slate (8 squares), orange (10 squares), sea green (14 squares), mint (25 squares), pale yellow (29 squares). An even split would hand every color 10 squares, so the indigo region is running 9 squares under that average while the pale yellow region is the roomiest area on the grid with 29 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 1 square while pale yellow has 29 to spare, a gap of 28. Attack the cramped end and the roomy regions tend to resolve themselves.

Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; pale yellow is the loosest, occupying only 45% of its 8×8 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

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 easy board.

Indigo: 1 square across rows 1–1, columns 7–7. It never leaves row 1, so its queen is locked to that row.

Red covers 1 square, spanning 1 row and 1 column (rows 6–6, columns 1–1). It never leaves row 6, so its queen is locked to that row.

Khaki occupies 1 square within rows 8–8, columns 5–5. It never leaves row 8, so its queen is locked to that row.

Blue: 4 squares across rows 1–3, columns 9–10.

Amber covers 7 squares, spanning 4 rows and 3 columns (rows 1–4, columns 1–3).

Slate occupies 8 squares within rows 5–9, columns 8–10.

Orange: 10 squares across rows 7–10, columns 6–10.

Sea green covers 14 squares, spanning 2 rows and 8 columns (rows 9–10, columns 1–8).

Mint occupies 25 squares within rows 1–5, columns 4–10.

Pale yellow: 29 squares across rows 2–9, columns 1–8.

Where to start on this board

Indigo is the natural opening on #96: it is the smallest region on the grid at 1 square, boxed into rows 1–1 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 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. Red, khaki 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 #96

A quick way to find the next move is to count how many different colors touch each line. On this board, row 10 is the narrowest, touched by only 2 colors. Columns 2, 3 and 4 are the tightest columns, with 3 colors.

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 9, meeting 4 colors, is the opposite case — the busiest line on #96 and the last one you should expect to crack.

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 #96 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 #96 is rated easy

The easy label comes out of four measurable things about #96: six of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 28. 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 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.

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 10 on this board.

Here is a shape check that gives nothing away: when #96 is solved, 4 queens sit on the outer edge of the grid and 6 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.