9x9 Queens Puzzle #107

About 9x9 Queens puzzle #107

9x9 puzzle #107 is rated hard. Nine colored regions cover its 81 squares, and your job is to seat nine queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

Puzzle #107 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 #107 are: amber (2 squares), khaki (2 squares), slate (6 squares), indigo (7 squares), orange (9 squares), mint (11 squares), pale yellow (12 squares), blue (13 squares), red (19 squares). An even split would hand every color 9 squares, so the amber region is running 7 squares under that average while the red region is the roomiest area on the grid with 19 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Amber has only 2 places to put its queen; red has 19. The spread between them on this board is 17 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Shape matters as much as size. The amber region is the most compact color here, filling 100% of the 2×1 box that encloses it, while slate is the most scattered at 50% of its 3×4 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 #107 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.

Amber occupies 2 squares within rows 1–2, columns 9–9. It sits wholly inside column 9, so its queen must be in that column.

Khaki: 2 squares across rows 7–8, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Slate covers 6 squares, spanning 3 rows and 4 columns (rows 3–5, columns 2–5).

Indigo occupies 7 squares within rows 1–3, columns 5–8.

Orange: 9 squares across rows 7–9, columns 1–5.

Mint covers 11 squares, spanning 5 rows and 4 columns (rows 1–5, columns 1–4).

Pale yellow occupies 12 squares within rows 5–8, columns 1–5.

Blue: 13 squares across rows 2–5, columns 4–9.

Red covers 19 squares, spanning 5 rows and 4 columns (rows 5–9, columns 6–9).

Where to start on this board

Amber is the natural opening on #107: it is the smallest region on the grid at 2 squares, boxed into rows 1–2 and columns 9–9. Since one of those squares must hold a queen, that block of 2 rows and 1 column is the most contested part of the board.

Because amber stays inside column 9, its queen belongs to that column and no other region may use it. Cross out the rest of column 9 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.

Work outward to the other flat regions next. Khaki 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 #107

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 6 and 9 are the narrowest, touched by only 2 colors. Columns 6, 7, 8 and 9 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 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 8 rows. By contrast row 5 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 81 squares of this grid. First: has any of the nine regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 9x9 boards almost always run one of those checks and forget the other.

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 #107 is rated hard

What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 2-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 17. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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 9x9

At 9x9 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.

One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #107 the place to look for one is around row 6.

As a sanity check rather than a spoiler: the finished solution to #107 puts 4 queens on the outer border and 5 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.