7x7 Queens Puzzle #261

About 7x7 Queens puzzle #261

Puzzle #261 is one of the hard-rated boards in the 7x7 set. Seven colors divide its 49 squares, and you win by placing seven queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.

As with every puzzle in this collection, #261 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 #261 are: pale yellow (1 square), indigo (2 squares), blue (5 squares), mint (6 squares), red (8 squares), amber (9 squares), slate (18 squares). An even split would hand every color 7 squares, so the pale yellow region is running 6 squares under that average while the slate region is the roomiest area on the grid with 18 squares.

Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So pale yellow, with 1 square, is under far more pressure than slate with 18. A 17-square gap like this one rewards players who start where the choices are fewest.

Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 2×1 frame; amber is the loosest, occupying only 60% of its 3×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

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: 1 square across rows 7–7, columns 7–7. It never leaves row 7, so its queen is locked to that row.

Indigo covers 2 squares, spanning 2 rows and 1 column (rows 1–2, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Blue occupies 5 squares within rows 1–4, columns 1–2.

Mint: 6 squares across rows 2–5, columns 6–7.

Red covers 8 squares, spanning 4 rows and 3 columns (rows 4–7, columns 5–7).

Amber occupies 9 squares within rows 1–3, columns 3–7.

Slate: 18 squares across rows 2–7, columns 1–4.

Where to start on this board

Pale yellow is the natural opening on #261: it is the smallest region on the grid at 1 square, boxed into rows 7–7 and columns 7–7. 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 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 is no more than two rows tall, so its queen is 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 #261

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

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 6 rows. Row 2, meeting 5 colors, is the opposite case — the busiest line on #261 and the last one you should expect to crack.

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.

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 #261 is rated hard

Ratings track how generous the opening is. On #261 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 17-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.

The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a hard board from the rest.

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 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.

Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #261, start hunting for it near row 6.

Here is a shape check that gives nothing away: when #261 is solved, 2 queens sit on the outer edge of the grid and 5 sit inside it, with 2 queens in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.