9x9 Queens Puzzle #296

About 9x9 Queens puzzle #296

This is puzzle number 296 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.

As with every puzzle in this collection, #296 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 #296 are: khaki (2 squares), orange (2 squares), slate (3 squares), indigo (5 squares), amber (6 squares), blue (11 squares), mint (15 squares), pale yellow (17 squares), red (20 squares). An even split would hand every color 9 squares, so the khaki region is running 7 squares under that average while the red 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. Khaki is working with 2 squares while red has 20 to spare, a gap of 18. Attack the cramped end and the roomy regions tend to resolve themselves.

Square count is only half the story — how a color is arranged counts too. Slate is the tidiest shape on #296, packing 100% of its 3×1 bounding box, whereas mint straggles across its 4×8 box at just 47%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Here is the full footprint of every color on puzzle #296, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Khaki occupies 2 squares within rows 8–9, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Orange: 2 squares across rows 9–9, columns 2–3. It never leaves row 9, so its queen is locked to that row.

Slate covers 3 squares, spanning 3 rows and 1 column (rows 3–5, columns 9–9). It sits wholly inside column 9, so its queen must be in that column.

Indigo occupies 5 squares within rows 1–2, columns 1–3.

Amber: 6 squares across rows 1–3, columns 3–5.

Blue covers 11 squares, spanning 3 rows and 5 columns (rows 1–3, columns 5–9).

Mint occupies 15 squares within rows 3–6, columns 1–8.

Pale yellow: 17 squares across rows 4–9, columns 1–5.

Red covers 20 squares, spanning 5 rows and 5 columns (rows 5–9, columns 5–9).

Where to start on this board

Begin where the choices are fewest. On this board that means khaki, which owns just 2 squares inside a 2×1 window running from row 8 to row 9 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 khaki 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. Orange 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 #296

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

The whole middle game reduces to two sweeps. Sweep the nine colors looking for one with a single square left; sweep the nine rows and nine columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 9x9 solve from a stalled one.

Resist the plausible-looking queen. On #296 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 #296 is rated hard

The rating reflects how much the board concedes up front. Puzzle #296 has three regions confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 18 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a hard board from the rest.

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.

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 7 on this board.

As a sanity check rather than a spoiler: the finished solution to #296 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.