8x8 Queens Puzzle #284

About 8x8 Queens puzzle #284

This is puzzle number 284 in the 8x8 collection, rated easy. The grid holds 64 squares split between eight colored regions, and a finished board carries eight queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

There is exactly one way to finish #284, 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 eight regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #284 are: pale yellow (1 square), indigo (2 squares), red (3 squares), khaki (4 squares), amber (5 squares), blue (15 squares), mint (17 squares), slate (17 squares). An even split would hand every color 8 squares, so the pale yellow region is running 7 squares under that average while the slate region is the roomiest area on the grid with 17 squares.

Every color takes one queen and one only, whatever its size — which is why the 16 square difference between pale yellow and slate is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.

Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×2 box that encloses it, while mint is the most scattered at 57% of its 6×5 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

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.

Pale yellow occupies 1 square within rows 7–7, columns 8–8. It never leaves row 7, so its queen is locked to that row.

Indigo: 2 squares across rows 1–1, columns 4–5. It never leaves row 1, so its queen is locked to that row.

Red covers 3 squares, spanning 3 rows and 1 column (rows 6–8, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Khaki occupies 4 squares within rows 7–8, columns 2–3.

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

Blue covers 15 squares, spanning 5 rows and 4 columns (rows 1–5, columns 5–8).

Mint occupies 17 squares within rows 1–6, columns 1–5.

Slate: 17 squares across rows 4–8, columns 4–8.

Where to start on this board

The most constrained color is where a solve should begin, and on #284 that is pale yellow with 1 square. It reaches from row 7 to row 7 and from column 8 to column 8 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.

Because pale yellow never leaves row 7, its queen must be in that row — and that instantly settles row 7 for everyone else. Sweep along it and X out every square that is not pale yellow. 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 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 #284

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 3, 4, 5, 6 and 8 are the narrowest, touched by only 3 colors. Columns 3, 6 and 7 are the tightest columns, with 2 colors.

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 7 rows. Row 7, meeting 4 colors, is the opposite case — the busiest line on #284 and the last one you should expect to crack.

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.

Two checks, run alternately, will carry you through the 64 squares of this grid. First: has any of the eight regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 8x8 boards almost always run one of those checks and forget the other.

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 #284 is rated easy

Ratings track how generous the opening is. On #284 you get four regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 16-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.

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.

Stuck on #284 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 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.

There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #284 is column 3.

Here is a shape check that gives nothing away: when #284 is solved, 4 queens sit on the outer edge of the grid and 4 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.