10x10 Queens Puzzle #165

About 10x10 Queens puzzle #165

Puzzle #165 is one of the medium-rated boards in the 10x10 set. Ten colors divide its 100 squares, and you win by placing ten queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.

There is exactly one way to finish #165, 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 ten regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #165 are: slate (1 square), red (3 squares), indigo (4 squares), blue (5 squares), pale yellow (5 squares), sea green (7 squares), orange (13 squares), mint (19 squares), amber (21 squares), khaki (22 squares). An even split would hand every color 10 squares, so the slate region is running 9 squares under that average while the khaki region is the roomiest area on the grid with 22 squares.

Every color takes one queen and one only, whatever its size — which is why the 21 square difference between slate and khaki 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.

Shape matters as much as size. The slate region is the most compact color here, filling 100% of the 1×1 box that encloses it, while orange is the most scattered at 43% of its 6×5 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.

Region by region

Here is the full footprint of every color on puzzle #165, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Slate: 1 square across rows 5–5, columns 3–3. It never leaves row 5, so its queen is locked to that row.

Red covers 3 squares, spanning 2 rows and 2 columns (rows 5–6, columns 9–10).

Indigo occupies 4 squares within rows 1–2, columns 8–10.

Blue: 5 squares across rows 2–4, columns 8–10.

Pale yellow covers 5 squares, spanning 3 rows and 2 columns (rows 5–7, columns 1–2).

Sea green occupies 7 squares within rows 7–10, columns 1–2.

Orange: 13 squares across rows 5–10, columns 3–7.

Mint covers 19 squares, spanning 8 rows and 5 columns (rows 1–8, columns 5–9).

Amber occupies 21 squares within rows 1–4, columns 1–6.

Khaki: 22 squares across rows 6–10, columns 4–10.

Where to start on this board

Begin where the choices are fewest. On this board that means slate, which owns just 1 square inside a 1×1 window running from row 5 to row 5 and column 3 to column 3. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

Because slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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 #165

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 3, 4, 9 and 10 are the narrowest, touched by only 3 colors. Columns 1, 2, 3, 4 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 3 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 7, meeting 5 colors, is the opposite case — the busiest line on #165 and the last one you should expect to crack.

Working through the middle game

The middle of #165 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 100 squares of this grid. First: has any of the ten regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 10x10 boards almost always run one of those checks and forget the other.

A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.

Why puzzle #165 is rated medium

The rating reflects how much the board concedes up front. Puzzle #165 has two regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 21 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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 medium board from the rest.

Stuck on #165 despite the medium 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 10x10

At 10x10 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 #165 is row 1.

Here is a shape check that gives nothing away: when #165 is solved, 3 queens sit on the outer edge of the grid and 7 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.