9x9 Queens Puzzle #283

About 9x9 Queens puzzle #283

9x9 puzzle #283 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.

There is exactly one way to finish #283, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these nine regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #283 are: pale yellow (3 squares), orange (5 squares), indigo (7 squares), amber (8 squares), mint (8 squares), khaki (9 squares), slate (11 squares), red (11 squares), blue (19 squares). An even split would hand every color 9 squares, so the pale yellow region is running 6 squares under that average while the blue region is the roomiest area on the grid with 19 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Pale yellow has only 3 places to put its queen; blue has 19. The spread between them on this board is 16 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 pale yellow region is the most compact color here, filling 100% of the 3×1 box that encloses it, while red is the most scattered at 55% of its 5×4 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

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 easy board.

Pale yellow covers 3 squares, spanning 3 rows and 1 column (rows 7–9, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Orange occupies 5 squares within rows 7–9, columns 2–3.

Indigo: 7 squares across rows 1–3, columns 5–8.

Amber covers 8 squares, spanning 4 rows and 3 columns (rows 1–4, columns 7–9).

Mint occupies 8 squares within rows 4–6, columns 6–9.

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

Slate covers 11 squares, spanning 4 rows and 5 columns (rows 4–7, columns 1–5).

Red occupies 11 squares within rows 5–9, columns 4–7.

Blue: 19 squares across rows 1–5, columns 1–5.

Where to start on this board

Pale yellow is the natural opening on #283: it is the smallest region on the grid at 3 squares, boxed into rows 7–9 and columns 1–1. Since one of those squares must hold a queen, that block of 3 rows and 1 column is the most contested part of the board.

Because pale yellow 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.

No region here is pinned to a shallow band of rows, so turn the same idea sideways: orange 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 #283

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3 and 6 are the narrowest, touched by only 3 colors. Columns 1, 2, 3, 4 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 3 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.

Resist the plausible-looking queen. On #283 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 #283 is rated easy

The easy label comes out of four measurable things about #283: one of its regions is locked to one row or column, its smallest color spans 3 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 16. Boards that score tight on those measures give up their first queen fast.

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.

Stuck on #283 despite the easy rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.

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 #283 the place to look for one is around row 1.

As a sanity check rather than a spoiler: the finished solution to #283 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.