11x11 Queens Puzzle #4

About 11x11 Queens puzzle #4

This is puzzle number 4 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 #4, 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 #4 are: amber (2 squares), teal (3 squares), sea green (5 squares), pale yellow (8 squares), indigo (13 squares), slate (14 squares), red (14 squares), mint (15 squares), orange (15 squares), blue (16 squares), khaki (16 squares). An even split would hand every color 11 squares, so the amber region is running 9 squares under that average while the khaki region is the roomiest area on the grid with 16 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 2 squares while khaki has 16 to spare, a gap of 14. Attack the cramped end and the roomy regions tend to resolve themselves.

A region's outline can constrain it more than its size does. Here amber is the most solid, filling 100% of its 1×2 box, and red the most diffuse at 39% of its 6×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

Region by region

The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.

Amber covers 2 squares, spanning 1 row and 2 columns (rows 2–2, columns 10–11). It never leaves row 2, so its queen is locked to that row.

Teal occupies 3 squares within rows 10–11, columns 5–6.

Sea green: 5 squares across rows 9–11, columns 9–11.

Pale yellow covers 8 squares, spanning 5 rows and 3 columns (rows 6–10, columns 9–11).

Indigo occupies 13 squares within rows 1–4, columns 2–7.

Slate: 14 squares across rows 3–8, columns 4–9.

Red covers 14 squares, spanning 6 rows and 6 columns (rows 3–8, columns 1–6).

Mint occupies 15 squares within rows 1–6, columns 1–3.

Orange: 15 squares across rows 6–11, columns 6–10.

Blue covers 16 squares, spanning 5 rows and 5 columns (rows 1–5, columns 7–11).

Khaki occupies 16 squares within rows 8–11, columns 1–6.

Where to start on this board

Open on the tightest color. Here that is amber, confined to 2 squares between rows 2 and 2 and columns 10 and 11. Its queen has nowhere else to go, which means those 1 row and 2 columns carry more constraint than anywhere else on #4.

Because amber never leaves row 2, its queen must be in that row — and that instantly settles row 2 for everyone else. Sweep along it and X out every square that is not amber. 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. Teal 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 #4

A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 3 colors. Columns 1 and 8 are the tightest columns, with 3 colors.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 10 rows. By contrast row 10 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

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.

Two checks, run alternately, will carry you through the 121 squares of this grid. First: has any of the eleven regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 11x11 boards almost always run one of those checks and forget the other.

A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.

Why puzzle #4 is rated medium

Ratings track how generous the opening is. On #4 you get one region pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 3 colors, and a 14-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.

The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a medium board from the rest.

Stuck on #4 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.

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 1 on this board.

As a sanity check rather than a spoiler: the finished solution to #4 puts 4 queens on the outer border and 7 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.