9x9 Queens Puzzle #297

About 9x9 Queens puzzle #297

Puzzle #297 is one of the medium-rated boards in the 9x9 set. Nine colors divide its 81 squares, and you win by placing nine queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.

As with every puzzle in this collection, #297 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.

How the colors divide this board

From smallest to largest, the regions on puzzle #297 are: amber (2 squares), blue (2 squares), pale yellow (2 squares), khaki (3 squares), orange (6 squares), indigo (7 squares), red (10 squares), slate (16 squares), mint (33 squares). An even split would hand every color 9 squares, so the amber region is running 7 squares under that average while the mint region is the roomiest area on the grid with 33 squares.

Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So amber, with 2 squares, is under far more pressure than mint with 33. A 31-square gap like this one rewards players who start where the choices are fewest.

Look at arrangement as well as area. Amber is the densest region on this grid at 100% of a 2×1 frame; mint is the loosest, occupying only 46% of its 8×9 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

Region by region

Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a medium board.

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

Blue covers 2 squares, spanning 1 row and 2 columns (rows 3–3, columns 6–7). It never leaves row 3, so its queen is locked to that row.

Pale yellow occupies 2 squares within rows 7–8, columns 9–9. It sits wholly inside column 9, so its queen must be in that column.

Khaki: 3 squares across rows 7–8, columns 7–8.

Orange covers 6 squares, spanning 3 rows and 4 columns (rows 7–9, columns 1–4).

Indigo occupies 7 squares within rows 1–2, columns 2–6.

Red: 10 squares across rows 4–8, columns 1–3.

Slate covers 16 squares, spanning 7 rows and 4 columns (rows 1–7, columns 6–9).

Mint occupies 33 squares within rows 2–9, columns 1–9.

Where to start on this board

The most constrained color is where a solve should begin, and on #297 that is amber with 2 squares. It reaches from row 1 to row 2 and from column 1 to column 1 — 2 rows by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.

Because amber stays inside column 1, its queen belongs to that column and no other region may use it. Cross out the rest of column 1 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. Blue, indigo, khaki and pale yellow 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 #297

A quick way to find the next move is to count how many different colors touch each line. On this board, row 9 is the narrowest, touched by only 2 colors. Column 5 is the tightest column, with 2 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 7 meets 6 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.

Keep alternating between two views of the board. In the region view you ask which of the nine colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 9x9 board unwinds.

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 #297 is rated medium

The medium label comes out of four measurable things about #297: three of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 2 colors, and its largest and smallest regions differ by 31. Boards that score tight on those measures give up their first queen fast.

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.

A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.

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.

Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #297, start hunting for it near row 9.

As a sanity check rather than a spoiler: the finished solution to #297 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.