9x9 Queens Puzzle #34

About 9x9 Queens puzzle #34

Board #34 of the 9x9 bank carries a hard rating. Its 81 squares are shared between nine colored regions, and the solution places nine queens: one per row, one per column, one per region, none of them side by side or corner to corner.

As with every puzzle in this collection, #34 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 #34 are: blue (1 square), indigo (5 squares), red (7 squares), mint (8 squares), khaki (9 squares), orange (9 squares), amber (10 squares), pale yellow (14 squares), slate (18 squares). An even split would hand every color 9 squares, so the blue region is running 8 squares under that average while the slate region is the roomiest area on the grid with 18 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Blue has only 1 place to put its queen; slate has 18. 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 blue region is the most compact color here, filling 100% of the 1×1 box that encloses it, while orange is the most scattered at 38% of its 4×6 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

Here is the full footprint of every color on puzzle #34, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

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

Indigo occupies 5 squares within rows 1–3, columns 8–9.

Red: 7 squares across rows 4–7, columns 3–5.

Mint covers 8 squares, spanning 3 rows and 5 columns (rows 3–5, columns 5–9).

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

Orange: 9 squares across rows 6–9, columns 1–6.

Amber covers 10 squares, spanning 3 rows and 7 columns (rows 1–3, columns 2–8).

Pale yellow occupies 14 squares within rows 5–8, columns 2–8.

Slate: 18 squares across rows 1–7, columns 1–6.

Where to start on this board

Open on the tightest color. Here that is blue, confined to 1 square between rows 3 and 3 and columns 7 and 7. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #34.

Because blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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.

No region here is pinned to a shallow band of rows, so turn the same idea sideways: indigo spans only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.

The tightest lines on puzzle #34

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 1 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 5 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

Working through the middle game

After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.

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.

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 #34 is rated hard

The rating reflects how much the board concedes up front. Puzzle #34 has two regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 17 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

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.

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 #34 the place to look for one is around row 9.

As a sanity check rather than a spoiler: the finished solution to #34 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.