9x9 Queens Puzzle #164

About 9x9 Queens puzzle #164

This is puzzle number 164 in the 9x9 collection, rated easy. The grid holds 81 squares split between nine colored regions, and a finished board carries nine 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 #164, 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 nine regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #164 are: mint (1 square), indigo (2 squares), orange (2 squares), khaki (3 squares), red (8 squares), slate (13 squares), blue (16 squares), amber (18 squares), pale yellow (18 squares). An even split would hand every color 9 squares, so the mint region is running 8 squares under that average while the pale yellow 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 17 square difference between mint and pale yellow 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 #164, packing 100% of its 2×1 bounding box, whereas slate straggles across its 6×5 box at just 43%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Below is every color on #164 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

Mint occupies 1 square within rows 4–4, columns 9–9. It never leaves row 4, so its queen is locked to that row.

Indigo: 2 squares across rows 1–2, columns 6–6. It sits wholly inside column 6, so its queen must be in that column.

Orange covers 2 squares, spanning 1 row and 2 columns (rows 9–9, columns 8–9). It never leaves row 9, so its queen is locked to that row.

Khaki occupies 3 squares within rows 8–9, columns 1–2.

Red: 8 squares across rows 5–9, columns 1–3.

Slate covers 13 squares, spanning 6 rows and 5 columns (rows 4–9, columns 3–7).

Blue occupies 16 squares within rows 1–4, columns 5–9.

Amber: 18 squares across rows 1–6, columns 1–4.

Pale yellow covers 18 squares, spanning 5 rows and 5 columns (rows 5–9, columns 5–9).

Where to start on this board

Begin where the choices are fewest. On this board that means mint, which owns just 1 square inside a 1×1 window running from row 4 to row 4 and column 9 to column 9. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

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.

Work outward to the other flat regions next. Orange, 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 #164

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

Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 colors, so one of that handful must take the row and the rest are squeezed into the other 8 rows. Row 9 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.

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.

Two checks, run alternately, will carry you through the 81 squares of this grid. First: has any of the nine regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 9x9 boards almost always run one of those checks and forget the other.

Resist the plausible-looking queen. On #164 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 #164 is rated easy

What earns a board its rating is the shape of its opening. Here that opening offers four single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 17. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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 #164 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 9x9

At 9x9 a full re-scan after every move starts to get slow, so work locally instead. Once a queen goes down, check only the regions, rows and columns that placement actually touched — that is where new information appeared. Fall back to a complete sweep only when the local pass turns up nothing, and your solve times will drop noticeably.

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 #164 is row 3.

Use this as a rough check on your progress: the completed board has 4 queens around the border and 5 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.