8x8 Queens Puzzle #90

About 8x8 Queens puzzle #90

Board #90 of the 8x8 bank carries a easy rating. Its 64 squares are shared between eight colored regions, and the solution places eight 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, #90 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 #90 are: indigo (1 square), amber (3 squares), khaki (3 squares), mint (5 squares), red (6 squares), slate (10 squares), pale yellow (10 squares), blue (26 squares). An even split would hand every color 8 squares, so the indigo region is running 7 squares under that average while the blue 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; blue 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.

Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; mint is the loosest, occupying only 56% of its 3×3 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

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.

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

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

Khaki occupies 3 squares within rows 8–8, columns 6–8. It never leaves row 8, so its queen is locked to that row.

Mint: 5 squares across rows 2–4, columns 1–3.

Red covers 6 squares, spanning 4 rows and 2 columns (rows 5–8, columns 1–2).

Slate occupies 10 squares within rows 2–7, columns 7–8.

Pale yellow: 10 squares across rows 6–8, columns 2–6.

Blue covers 26 squares, spanning 6 rows and 7 columns (rows 1–6, columns 1–7).

Where to start on this board

Indigo is the natural opening on #90: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 3–3. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.

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. Khaki and amber 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 #90

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 3, 4, 5, 7 and 8 are the narrowest, touched by only 3 colors. Columns 4 and 5 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 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 7 rows. By contrast row 6 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 eight 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 8x8 board unwinds.

Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.

Why puzzle #90 is rated easy

The easy label comes out of four measurable things about #90: three of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 25. 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.

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 8x8

At 8x8 the whole grid stays within a single glance, which makes systematic scanning practical. Read the rows in order and count the colors in each — a row touched by two colors is far more constrained than one touched by five, and that is usually where the next forced move hides. The board is small enough that re-scanning from the top after each placement costs almost nothing.

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 #90 the place to look for one is around column 4.

As a sanity check rather than a spoiler: the finished solution to #90 puts 4 queens on the outer border and 4 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.