9x9 Queens Puzzle #199

About 9x9 Queens puzzle #199

9x9 puzzle #199 is rated easy. Nine colored regions cover its 81 squares, and your job is to seat nine queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

Like every board on this site, #199 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.

How the colors divide this board

From smallest to largest, the regions on puzzle #199 are: orange (2 squares), amber (3 squares), red (3 squares), indigo (5 squares), blue (6 squares), khaki (6 squares), pale yellow (11 squares), mint (14 squares), slate (31 squares). An even split would hand every color 9 squares, so the orange region is running 7 squares under that average while the slate region is the roomiest area on the grid with 31 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Orange has only 2 places to put its queen; slate has 31. The spread between them on this board is 29 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #199, packing 100% of its 1×5 bounding box, whereas mint straggles across its 6×5 box at just 47%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Below is every color on #199 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

Orange covers 2 squares, spanning 1 row and 2 columns (rows 9–9, columns 8–9). It never leaves row 9, so its queen is locked to that row.

Amber occupies 3 squares within rows 1–2, columns 3–4.

Red: 3 squares across rows 6–7, columns 4–5.

Indigo covers 5 squares, spanning 1 row and 5 columns (rows 1–1, columns 5–9). It never leaves row 1, so its queen is locked to that row.

Blue occupies 6 squares within rows 1–3, columns 1–2.

Khaki: 6 squares across rows 8–9, columns 4–7.

Pale yellow covers 11 squares, spanning 4 rows and 4 columns (rows 6–9, columns 1–4).

Mint occupies 14 squares within rows 2–7, columns 1–5.

Slate: 31 squares across rows 2–8, columns 3–9.

Where to start on this board

Orange is the natural opening on #199: it is the smallest region on the grid at 2 squares, boxed into rows 9–9 and columns 8–9. Since one of those squares must hold a queen, that block of 1 row and 2 columns is the most contested part of the board.

Because orange never leaves row 9, its queen must be in that row — and that instantly settles row 9 for everyone else. Sweep along it and X out every square that is not orange. 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. Indigo, amber, red and khaki 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 #199

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 4 and 5 are the narrowest, touched by only 2 colors. Columns 1, 2, 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 7 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.

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 #199 is rated easy

What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 2-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 29. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

If #199 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.

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 #199 the place to look for one is around row 4.

As a sanity check rather than a spoiler: the finished solution to #199 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.