10x10 Queens Puzzle #2

About 10x10 Queens puzzle #2

Board #2 of the 10x10 bank carries a easy 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.

Puzzle #2 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 #2 are: sea green (1 square), indigo (3 squares), mint (3 squares), khaki (5 squares), red (6 squares), orange (6 squares), pale yellow (9 squares), amber (15 squares), slate (18 squares), blue (34 squares). An even split would hand every color 10 squares, so the sea green region is running 9 squares under that average while the blue region is the roomiest area on the grid with 34 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Sea green has only 1 place to put its queen; blue has 34. The spread between them on this board is 33 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #2, packing 100% of its 1×3 bounding box, whereas amber straggles across its 4×7 box at just 54%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

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 easy board.

Sea green occupies 1 square within rows 10–10, columns 9–9. It never leaves row 10, so its queen is locked to that row.

Indigo: 3 squares across rows 1–1, columns 8–10. It never leaves row 1, so its queen is locked to that row.

Mint covers 3 squares, spanning 3 rows and 1 column (rows 3–5, columns 4–4). It sits wholly inside column 4, so its queen must be in that column.

Khaki occupies 5 squares within rows 8–10, columns 5–6.

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

Orange covers 6 squares, spanning 2 rows and 4 columns (rows 9–10, columns 1–4).

Pale yellow occupies 9 squares within rows 7–10, columns 8–10.

Amber: 15 squares across rows 1–4, columns 1–7.

Slate covers 18 squares, spanning 8 rows and 4 columns (rows 2–9, columns 1–4).

Blue occupies 34 squares within rows 2–10, columns 4–10.

Where to start on this board

The most constrained color is where a solve should begin, and on #2 that is sea green with 1 square. It reaches from row 10 to row 10 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 sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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, red and orange 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 #2

A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Column 7 is the tightest column, with 2 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 10 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

After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.

The whole middle game reduces to two sweeps. Sweep the ten colors looking for one with a single square left; sweep the ten rows and ten columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 10x10 solve from a stalled one.

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 #2 is rated easy

The easy label comes out of four measurable things about #2: 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 33. Boards that score tight on those measures give up their first queen fast.

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 #2 the place to look for one is around row 1.

As a sanity check rather than a spoiler: the finished solution to #2 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.