9x9 Queens Puzzle #194

About 9x9 Queens puzzle #194

Board #194 of the 9x9 bank carries a medium 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, #194 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 #194 are: indigo (1 square), orange (1 square), mint (2 squares), slate (5 squares), khaki (7 squares), amber (10 squares), pale yellow (10 squares), blue (19 squares), red (26 squares). An even split would hand every color 9 squares, so the indigo region is running 8 squares under that average while the red region is the roomiest area on the grid with 26 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 1 place to put its queen; red has 26. The spread between them on this board is 25 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and pale yellow the most diffuse at 56% of its 6×3 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

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.

Orange: 1 square across rows 9–9, columns 3–3. It never leaves row 9, so its queen is locked to that row.

Mint covers 2 squares, spanning 2 rows and 1 column (rows 4–5, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Slate occupies 5 squares within rows 3–6, columns 8–9.

Khaki: 7 squares across rows 6–9, columns 7–9.

Amber covers 10 squares, spanning 4 rows and 3 columns (rows 1–4, columns 1–3).

Pale yellow occupies 10 squares within rows 3–8, columns 6–8.

Blue: 19 squares across rows 1–4, columns 4–9.

Red covers 26 squares, spanning 6 rows and 6 columns (rows 4–9, columns 1–6).

Where to start on this board

Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 5 and 5. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #194.

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. Orange and mint 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 #194

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

The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.

Why puzzle #194 is rated medium

Ratings track how generous the opening is. On #194 you get five regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 25-square gap between the largest and smallest regions. Those four numbers largely explain the medium 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.

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 #194 the place to look for one is around row 2.

As a sanity check rather than a spoiler: the finished solution to #194 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.