10x10 Queens Puzzle #204

About 10x10 Queens puzzle #204

This is puzzle number 204 in the 10x10 collection, rated medium. 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 #204 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 #204 are: indigo (1 square), khaki (2 squares), blue (6 squares), sea green (6 squares), mint (9 squares), amber (11 squares), pale yellow (14 squares), orange (14 squares), slate (16 squares), red (21 squares). An even split would hand every color 10 squares, so the indigo region is running 9 squares under that average while the red region is the roomiest area on the grid with 21 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 1 place to put its queen; red has 21. The spread between them on this board is 20 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 indigo is the most solid, filling 100% of its 1×1 box, and mint the most diffuse at 43% of its 3×7 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

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 medium board.

Indigo: 1 square across rows 1–1, columns 10–10. It never leaves row 1, so its queen is locked to that row.

Khaki covers 2 squares, spanning 2 rows and 1 column (rows 8–9, columns 3–3). It sits wholly inside column 3, so its queen must be in that column.

Blue occupies 6 squares within rows 1–3, columns 1–3.

Sea green: 6 squares across rows 8–10, columns 1–3.

Mint covers 9 squares, spanning 3 rows and 7 columns (rows 2–4, columns 4–10).

Amber occupies 11 squares within rows 1–2, columns 3–9.

Pale yellow: 14 squares across rows 5–8, columns 1–6.

Orange covers 14 squares, spanning 3 rows and 7 columns (rows 8–10, columns 4–10).

Slate occupies 16 squares within rows 3–6, columns 1–7.

Red: 21 squares across rows 3–9, columns 6–10.

Where to start on this board

Indigo is the natural opening on #204: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 10–10. 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 indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Amber 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 #204

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 7 and 10 are the narrowest, touched by only 2 colors. Columns 1, 2, 8, 9 and 10 are the tightest columns, with 4 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 8, meeting 5 colors, is the opposite case — the busiest line on #204 and the last one you should expect to crack.

Working through the middle game

Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.

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 #204 is rated medium

The medium label comes out of four measurable things about #204: three 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 20. 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.

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 #204 the place to look for one is around row 7.

Here is a shape check that gives nothing away: when #204 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.