7x7 Queens Puzzle #283

About 7x7 Queens puzzle #283

7x7 puzzle #283 is rated easy. Seven colored regions cover its 49 squares, and your job is to seat seven 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 seven regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #283 are: mint (1 square), blue (3 squares), slate (4 squares), pale yellow (5 squares), indigo (7 squares), amber (14 squares), red (15 squares). An even split would hand every color 7 squares, so the mint region is running 6 squares under that average while the red region is the roomiest area on the grid with 15 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Mint is working with 1 square while red has 15 to spare, a gap of 14. Attack the cramped end and the roomy regions tend to resolve themselves.

Look at arrangement as well as area. Blue is the densest region on this grid at 100% of a 3×1 frame; pale yellow 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.

Mint covers 1 square, spanning 1 row and 1 column (rows 4–4, columns 3–3). It never leaves row 4, so its queen is locked to that row.

Blue occupies 3 squares within rows 3–5, columns 7–7. It sits wholly inside column 7, so its queen must be in that column.

Slate: 4 squares across rows 4–6, columns 1–2.

Pale yellow covers 5 squares, spanning 3 rows and 3 columns (rows 5–7, columns 1–3).

Indigo occupies 7 squares within rows 1–4, columns 5–7.

Amber: 14 squares across rows 1–3, columns 1–5.

Red covers 15 squares, spanning 4 rows and 5 columns (rows 4–7, columns 3–7).

Where to start on this board

The most constrained color is where a solve should begin, and on #283 that is mint with 1 square. It reaches from row 4 to row 4 and from column 3 to column 3 — 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 mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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.

No region here is pinned to a shallow band of rows, so turn the same idea sideways: slate 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 and 7 are the narrowest, touched by only 2 colors. Columns 4 and 6 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 6 rows. By contrast row 4 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

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.

With seven queens to seat and seven regions competing over 49 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 7x7 boards quickly and players who stall halfway.

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 #283 is rated easy

The easy label comes out of four measurable things about #283: three of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 14. Boards that score tight on those measures give up their first queen fast.

Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.

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 7x7

At 7x7 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.

Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 1 on this board.

As a sanity check rather than a spoiler: the finished solution to #283 puts 4 queens on the outer border and 3 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.