7x7 Queens Puzzle #161

About 7x7 Queens puzzle #161

Puzzle #161 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, #161 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 #161 are: blue (3 squares), red (3 squares), indigo (5 squares), pale yellow (5 squares), amber (6 squares), mint (9 squares), slate (18 squares). An even split would hand every color 7 squares, so the blue region is running 4 squares under that average while the slate region is the roomiest area on the grid with 18 squares.

Every color takes one queen and one only, whatever its size — which is why the 15 square difference between blue 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. Amber is the tidiest shape on #161, packing 100% of its 2×3 bounding box, whereas mint straggles across its 3×6 box at just 50%. 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.

Blue occupies 3 squares within rows 1–3, columns 7–7. It sits wholly inside column 7, so its queen must be in that column.

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

Indigo covers 5 squares, spanning 2 rows and 3 columns (rows 1–2, columns 1–3).

Pale yellow occupies 5 squares within rows 7–7, columns 2–6. It never leaves row 7, so its queen is locked to that row.

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

Mint covers 9 squares, spanning 3 rows and 6 columns (rows 2–4, columns 1–6).

Slate occupies 18 squares within rows 4–7, columns 1–7.

Where to start on this board

Blue is the natural opening on #161: it is the smallest region on the grid at 3 squares, boxed into rows 1–3 and columns 7–7. Since one of those squares must hold a queen, that block of 3 rows and 1 column is the most contested part of the board.

Because blue stays inside column 7, its queen belongs to that column and no other region may use it. Cross out the rest of column 7 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.

Work outward to the other flat regions next. Pale yellow, indigo, amber 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 #161

A quick way to find the next move is to count how many different colors touch each line. On this board, row 5 is the narrowest, touched by only 1 color. Columns 1 and 7 are the tightest columns, with 3 colors.

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 1 color has to be claimed by one of them, which pushes every remaining region into the other 6 rows. Row 2, meeting 4 colors, is the opposite case — the busiest line on #161 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.

Keep alternating between two views of the board. In the region view you ask which of the seven 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 7x7 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 #161 is rated hard

Ratings track how generous the opening is. On #161 you get two regions pinned to a single row or column, a smallest color of 3 squares, a tightest row crossed by 1 color, 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.

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.

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 #161 is row 5.

Here is a shape check that gives nothing away: when #161 is solved, 3 queens sit on the outer edge of the grid and 4 sit inside it, with 1 queen in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.