10x10 Queens Puzzle #228

About 10x10 Queens puzzle #228

This is puzzle number 228 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.

There is exactly one way to finish #228, 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 #228 are: amber (1 square), indigo (2 squares), pale yellow (2 squares), mint (8 squares), slate (8 squares), khaki (9 squares), sea green (9 squares), blue (14 squares), orange (18 squares), red (29 squares). An even split would hand every color 10 squares, so the amber region is running 9 squares under that average while the red region is the roomiest area on the grid with 29 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 1 square while red has 29 to spare, a gap of 28. Attack the cramped end and the roomy regions tend to resolve themselves.

A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 2×1 box, and red the most diffuse at 52% of its 7×8 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.

Amber: 1 square across rows 2–2, columns 4–4. It never leaves row 2, so its queen is locked to that row.

Indigo covers 2 squares, spanning 2 rows and 1 column (rows 1–2, columns 10–10). It sits wholly inside column 10, so its queen must be in that column.

Pale yellow occupies 2 squares within rows 7–7, columns 7–8. It never leaves row 7, so its queen is locked to that row.

Mint: 8 squares across rows 1–4, columns 1–3.

Slate covers 8 squares, spanning 3 rows and 4 columns (rows 3–5, columns 1–4).

Khaki occupies 9 squares within rows 6–10, columns 8–10.

Sea green: 9 squares across rows 7–10, columns 6–9.

Blue covers 14 squares, spanning 3 rows and 7 columns (rows 1–3, columns 3–9).

Orange occupies 18 squares within rows 6–10, columns 1–6.

Red: 29 squares across rows 3–9, columns 3–10.

Where to start on this board

Amber is the natural opening on #228: it is the smallest region on the grid at 1 square, boxed into rows 2–2 and columns 4–4. 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. Pale yellow and indigo 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 #228

A quick way to find the next move is to count how many different colors touch each line. On this board, row 5 is the narrowest, touched by only 2 colors. Columns 1, 2, 5 and 10 are the tightest columns, with 3 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 7, meeting 5 colors, is the opposite case — the busiest line on #228 and the last one you should expect to crack.

Working through the middle game

The middle of #228 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.

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 #228 is rated medium

The medium label comes out of four measurable things about #228: four 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 28. 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.

Stuck on #228 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.

Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 5 on this board.

Here is a shape check that gives nothing away: when #228 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.