7x7 Queens Puzzle #13

About 7x7 Queens puzzle #13

Puzzle #13 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.

Like every board on this site, #13 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.

How the colors divide this board

From smallest to largest, the regions on puzzle #13 are: pale yellow (2 squares), indigo (3 squares), red (3 squares), blue (4 squares), amber (9 squares), mint (11 squares), slate (17 squares). An even split would hand every color 7 squares, so the pale yellow region is running 5 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 15 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.

Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #13, packing 100% of its 1×3 bounding box, whereas amber straggles across its 3×5 box at just 60%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

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 covers 2 squares, spanning 1 row and 2 columns (rows 7–7, columns 6–7). It never leaves row 7, so its queen is locked to that row.

Indigo occupies 3 squares within rows 1–1, columns 1–3. It never leaves row 1, so its queen is locked to that row.

Red: 3 squares across rows 6–7, columns 4–5.

Blue covers 4 squares, spanning 3 rows and 2 columns (rows 2–4, columns 1–2).

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

Mint: 11 squares across rows 2–6, columns 5–7.

Slate covers 17 squares, spanning 5 rows and 4 columns (rows 3–7, columns 1–4).

Where to start on this board

Pale yellow is the natural opening on #13: it is the smallest region on the grid at 2 squares, boxed into rows 7–7 and columns 6–7. Since one of those squares must hold a queen, that block of 1 row and 2 columns 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 and red 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 #13

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 5 are the narrowest, touched by only 2 colors. Columns 1, 2, 3, 4, 5, 6 and 7 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 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 3 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

The middle of #13 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.

Two checks, run alternately, will carry you through the 49 squares of this grid. First: has any of the seven regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 7x7 boards almost always run one of those checks and forget the other.

Resist the plausible-looking queen. On #13 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 #13 is rated hard

Ratings track how generous the opening is. On #13 you get two regions pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 2 colors, and a 15-square gap between the largest and smallest regions. Those four numbers largely explain the hard 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.

If #13 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.

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.

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 #13 is row 1.

As a sanity check rather than a spoiler: the finished solution to #13 puts 3 queens on the outer border and 4 queens in the interior, 1 of them in an actual 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.