11x11 Queens Puzzle #96

About 11x11 Queens puzzle #96

This is puzzle number 96 in the 11x11 collection, rated medium. The grid holds 121 squares split between eleven colored regions, and a finished board carries eleven queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

There is exactly one way to finish #96, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these eleven regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #96 are: teal (1 square), orange (2 squares), indigo (4 squares), red (8 squares), amber (9 squares), slate (10 squares), khaki (10 squares), blue (12 squares), mint (14 squares), sea green (14 squares), pale yellow (37 squares). An even split would hand every color 11 squares, so the teal region is running 10 squares under that average while the pale yellow region is the roomiest area on the grid with 37 squares.

Every color takes one queen and one only, whatever its size — which is why the 36 square difference between teal and pale yellow 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.

Shape matters as much as size. The orange region is the most compact color here, filling 100% of the 1×2 box that encloses it, while pale yellow is the most scattered at 41% of its 10×9 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

Here is the full footprint of every color on puzzle #96, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Teal: 1 square across rows 11–11, columns 2–2. It never leaves row 11, so its queen is locked to that row.

Orange covers 2 squares, spanning 1 row and 2 columns (rows 9–9, columns 5–6). It never leaves row 9, so its queen is locked to that row.

Indigo occupies 4 squares within rows 1–3, columns 1–2.

Red: 8 squares across rows 6–8, columns 8–11.

Amber covers 9 squares, spanning 4 rows and 4 columns (rows 1–4, columns 8–11).

Slate occupies 10 squares within rows 2–6, columns 8–11.

Khaki: 10 squares across rows 8–11, columns 9–11.

Blue covers 12 squares, spanning 4 rows and 6 columns (rows 1–4, columns 3–8).

Mint occupies 14 squares within rows 2–6, columns 3–7.

Sea green: 14 squares across rows 9–11, columns 3–8.

Pale yellow covers 37 squares, spanning 10 rows and 9 columns (rows 2–11, columns 1–9).

Where to start on this board

Open on the tightest color. Here that is teal, confined to 1 square between rows 11 and 11 and columns 2 and 2. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #96.

Because teal never leaves row 11, its queen must be in that row — and that instantly settles row 11 for everyone else. Sweep along it and X out every square that is not teal. 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. Orange 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 #96

A quick way to find the next move is to count how many different colors touch each line. On this board, row 7 is the narrowest, touched by only 2 colors. Column 1 is the tightest column, with 2 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 10 rows. Row 3, meeting 6 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.

With eleven queens to seat and eleven regions competing over 121 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 11x11 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 #96 is rated medium

The rating reflects how much the board concedes up front. Puzzle #96 has three 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 36 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

Stuck on #96 despite the medium rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.

Techniques that work well at 11x11

At 11x11 you are managing 121 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.

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 #96 is row 7.

Here is a shape check that gives nothing away: when #96 is solved, 3 queens sit on the outer edge of the grid and 8 sit inside it, with 1 queen in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.