10x10 Queens Puzzle #162

About 10x10 Queens puzzle #162

Board #162 of the 10x10 bank carries a hard rating. Its 100 squares are shared between ten colored regions, and the solution places ten queens: one per row, one per column, one per region, none of them side by side or corner to corner.

As with every puzzle in this collection, #162 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 #162 are: amber (1 square), red (5 squares), sea green (5 squares), slate (6 squares), pale yellow (6 squares), indigo (10 squares), mint (10 squares), blue (13 squares), orange (13 squares), khaki (31 squares). An even split would hand every color 10 squares, so the amber region is running 9 squares under that average while the khaki region is the roomiest area on the grid with 31 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Amber has only 1 place to put its queen; khaki has 31. The spread between them on this board is 30 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Shape matters as much as size. The amber region is the most compact color here, filling 100% of the 1×1 box that encloses it, while khaki is the most scattered at 48% of its 8×8 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

Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a hard board.

Amber: 1 square across rows 2–2, columns 9–9. It never leaves row 2, so its queen is locked to that row.

Red covers 5 squares, spanning 3 rows and 3 columns (rows 4–6, columns 1–3).

Sea green occupies 5 squares within rows 9–10, columns 1–3.

Slate: 6 squares across rows 2–5, columns 9–10.

Pale yellow covers 6 squares, spanning 2 rows and 4 columns (rows 6–7, columns 1–4).

Indigo occupies 10 squares within rows 1–5, columns 7–10.

Mint: 10 squares across rows 1–5, columns 1–3.

Blue covers 13 squares, spanning 5 rows and 4 columns (rows 1–5, columns 3–6).

Orange occupies 13 squares within rows 6–10, columns 7–10.

Khaki: 31 squares across rows 3–10, columns 1–8.

Where to start on this board

The most constrained color is where a solve should begin, and on #162 that is amber with 1 square. It reaches from row 2 to row 2 and from column 9 to column 9 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.

Because amber never leaves row 2, its queen must be in that row — and that instantly settles row 2 for everyone else. Sweep along it and X out every square that is not amber. 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. Pale yellow and sea green 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 #162

A quick way to find the next move is to count how many different colors touch each line. On this board, row 8 is the narrowest, touched by only 2 colors. Columns 5 and 6 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 9 rows. Row 5, meeting 6 colors, is the opposite case — the busiest line on #162 and the last one you should expect to crack.

Working through the middle game

Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.

Keep alternating between two views of the board. In the region view you ask which of the ten 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 10x10 board unwinds.

The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.

Why puzzle #162 is rated hard

The hard label comes out of four measurable things about #162: two of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 30. Boards that score tight on those measures give up their first queen fast.

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 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.

One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #162 the place to look for one is around row 8.

Here is a shape check that gives nothing away: when #162 is solved, 4 queens sit on the outer edge of the grid and 6 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.