10x10 Queens Puzzle #268

About 10x10 Queens puzzle #268

This is puzzle number 268 in the 10x10 collection, rated easy. The grid holds 100 squares split between ten colored regions, and a finished board carries ten queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

Puzzle #268 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.

How the colors divide this board

From smallest to largest, the regions on puzzle #268 are: amber (1 square), orange (1 square), red (3 squares), indigo (4 squares), slate (8 squares), sea green (10 squares), khaki (16 squares), pale yellow (18 squares), blue (19 squares), mint (20 squares). An even split would hand every color 10 squares, so the amber region is running 9 squares under that average while the mint region is the roomiest area on the grid with 20 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; mint has 20. The spread between them on this board is 19 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

A region's outline can constrain it more than its size does. Here amber is the most solid, filling 100% of its 1×1 box, and pale yellow the most diffuse at 50% of its 4×9 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

Region by region

Here is the full footprint of every color on puzzle #268, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Amber covers 1 square, spanning 1 row and 1 column (rows 2–2, columns 1–1). It never leaves row 2, so its queen is locked to that row.

Orange occupies 1 square within rows 9–9, columns 7–7. It never leaves row 9, so its queen is locked to that row.

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

Indigo covers 4 squares, spanning 2 rows and 3 columns (rows 1–2, columns 8–10).

Slate occupies 8 squares within rows 4–6, columns 1–3.

Sea green: 10 squares across rows 8–10, columns 6–10.

Khaki covers 16 squares, spanning 4 rows and 6 columns (rows 7–10, columns 1–6).

Pale yellow occupies 18 squares within rows 5–8, columns 2–10.

Blue: 19 squares across rows 1–5, columns 1–6.

Mint covers 20 squares, spanning 6 rows and 5 columns (rows 1–6, columns 6–10).

Where to start on this board

Amber is the natural opening on #268: it is the smallest region on the grid at 1 square, boxed into rows 2–2 and columns 1–1. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of 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. Orange, 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 #268

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3, 7 and 10 are the narrowest, touched by only 2 colors. Columns 1, 2, 3, 4, 5, 7, 8, 9 and 10 are the tightest columns, with 4 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 9 rows. By contrast row 5 meets 5 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

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.

With ten queens to seat and ten regions competing over 100 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 10x10 boards quickly and players who stall halfway.

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 #268 is rated easy

The rating reflects how much the board concedes up front. Puzzle #268 has four regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 19 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.

When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.

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 #268 the place to look for one is around row 3.

As a sanity check rather than a spoiler: the finished solution to #268 puts 3 queens on the outer border and 7 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.