9x9 Queens Puzzle #152

About 9x9 Queens puzzle #152

This is puzzle number 152 in the 9x9 collection, rated easy. 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 #152 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 #152 are: indigo (1 square), slate (1 square), orange (1 square), amber (7 squares), red (8 squares), pale yellow (12 squares), khaki (12 squares), blue (18 squares), mint (21 squares). An even split would hand every color 9 squares, so the indigo region is running 8 squares under that average while the mint region is the roomiest area on the grid with 21 squares.

Every color takes one queen and one only, whatever its size — which is why the 20 square difference between indigo and mint 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.

Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; red is the loosest, occupying only 40% of its 5×4 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.

Indigo occupies 1 square within rows 1–1, columns 5–5. It never leaves row 1, so its queen is locked to that row.

Slate: 1 square across rows 5–5, columns 9–9. It never leaves row 5, so its queen is locked to that row.

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

Amber occupies 7 squares within rows 1–2, columns 1–5.

Red: 8 squares across rows 5–9, columns 2–5.

Pale yellow covers 12 squares, spanning 5 rows and 3 columns (rows 5–9, columns 1–3).

Khaki occupies 12 squares within rows 7–9, columns 5–9.

Blue: 18 squares across rows 1–6, columns 6–9.

Mint covers 21 squares, spanning 7 rows and 7 columns (rows 2–8, columns 1–7).

Where to start on this board

Begin where the choices are fewest. On this board that means indigo, which owns just 1 square inside a 1×1 window running from row 1 to row 1 and column 5 to column 5. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Slate, orange and amber 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 #152

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 4 are the narrowest, touched by only 2 colors. Column 8 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 8 rows. Row 5, meeting 5 colors, is the opposite case — the busiest line on #152 and the last one you should expect to crack.

Working through the middle game

The middle of #152 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.

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 #152 is rated easy

Ratings track how generous the opening is. On #152 you get six regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 20-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.

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.

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 #152 is row 3.

Here is a shape check that gives nothing away: when #152 is solved, 3 queens sit on the outer edge of the grid and 6 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.