9x9 Queens Puzzle #56

About 9x9 Queens puzzle #56

This is puzzle number 56 in the 9x9 collection, rated hard. The grid holds 81 squares split between nine colored regions, and a finished board carries nine queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

Puzzle #56 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 #56 are: amber (1 square), red (2 squares), indigo (6 squares), mint (6 squares), orange (6 squares), pale yellow (12 squares), blue (14 squares), khaki (14 squares), slate (20 squares). An even split would hand every color 9 squares, so the amber region is running 8 squares under that average while the slate region is the roomiest area on the grid with 20 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 1 square while slate has 20 to spare, a gap of 19. Attack the cramped end and the roomy regions tend to resolve themselves.

Look at arrangement as well as area. Amber is the densest region on this grid at 100% of a 1×1 frame; pale yellow is the loosest, occupying only 48% of its 5×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

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 occupies 1 square within rows 2–2, columns 1–1. It never leaves row 2, so its queen is locked to that row.

Red: 2 squares across rows 5–6, columns 6–6. It sits wholly inside column 6, so its queen must be in that column.

Indigo covers 6 squares, spanning 3 rows and 4 columns (rows 1–3, columns 6–9).

Mint occupies 6 squares within rows 3–4, columns 6–9.

Orange: 6 squares across rows 8–9, columns 4–8.

Pale yellow covers 12 squares, spanning 5 rows and 5 columns (rows 5–9, columns 2–6).

Blue occupies 14 squares within rows 1–3, columns 1–8.

Khaki: 14 squares across rows 4–9, columns 7–9.

Slate covers 20 squares, spanning 7 rows and 5 columns (rows 3–9, columns 1–5).

Where to start on this board

Begin where the choices are fewest. On this board that means amber, which owns just 1 square inside a 1×1 window running from row 2 to row 2 and column 1 to column 1. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

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. Mint, red and orange 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 #56

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 2 colors. Columns 1, 2, 3 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 9 meets 4 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 nine queens to seat and nine regions competing over 81 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 9x9 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 #56 is rated hard

Ratings track how generous the opening is. On #56 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 19-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.

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.

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.

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 #56 puts 3 queens on the outer border and 6 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.