11x11 Queens Puzzle #112

About 11x11 Queens puzzle #112

This is puzzle number 112 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 #112, 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 #112 are: mint (1 square), indigo (2 squares), blue (2 squares), pale yellow (2 squares), teal (2 squares), amber (8 squares), sea green (8 squares), slate (17 squares), khaki (20 squares), red (23 squares), orange (36 squares). An even split would hand every color 11 squares, so the mint region is running 10 squares under that average while the orange region is the roomiest area on the grid with 36 squares.

Every color takes one queen and one only, whatever its size — which is why the 35 square difference between mint and orange 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 indigo region is the most compact color here, filling 100% of the 2×1 box that encloses it, while red is the most scattered at 43% of its 9×6 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

Below is every color on #112 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

Mint covers 1 square, spanning 1 row and 1 column (rows 4–4, columns 1–1). It never leaves row 4, so its queen is locked to that row.

Indigo occupies 2 squares within rows 1–2, columns 11–11. It sits wholly inside column 11, so its queen must be in that column.

Blue: 2 squares across rows 3–3, columns 10–11. It never leaves row 3, so its queen is locked to that row.

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

Teal occupies 2 squares within rows 11–11, columns 5–6. It never leaves row 11, so its queen is locked to that row.

Amber: 8 squares across rows 1–3, columns 8–10.

Sea green covers 8 squares, spanning 3 rows and 5 columns (rows 9–11, columns 5–9).

Slate occupies 17 squares within rows 1–7, columns 3–6.

Khaki: 20 squares across rows 4–11, columns 9–11.

Red covers 23 squares, spanning 9 rows and 6 columns (rows 1–9, columns 5–10).

Orange occupies 36 squares within rows 1–11, columns 1–6.

Where to start on this board

Open on the tightest color. Here that is mint, confined to 1 square between rows 4 and 4 and columns 1 and 1. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #112.

Because mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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. Blue, pale yellow, teal and indigo 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 #112

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 8 and 10 are the narrowest, touched by only 3 colors. Columns 2, 3, 4 and 7 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 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 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 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.

Resist the plausible-looking queen. On #112 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 #112 is rated medium

What earns a board its rating is the shape of its opening. Here that opening offers six single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 35. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

Stuck on #112 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 #112 is column 2.

As a sanity check rather than a spoiler: the finished solution to #112 puts 3 queens on the outer border and 8 queens in the interior, 1 of them in an actual 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.