10x10 Queens Puzzle #138

About 10x10 Queens puzzle #138

Board #138 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, #138 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 #138 are: indigo (1 square), sea green (1 square), amber (6 squares), pale yellow (6 squares), orange (7 squares), blue (10 squares), red (12 squares), khaki (17 squares), mint (18 squares), slate (22 squares). An even split would hand every color 10 squares, so the indigo region is running 9 squares under that average while the slate region is the roomiest area on the grid with 22 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; slate has 22. The spread between them on this board is 21 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 #138, packing 100% of its 1×1 bounding box, whereas red straggles across its 7×4 box at just 43%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Here is the full footprint of every color on puzzle #138, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

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

Sea green covers 1 square, spanning 1 row and 1 column (rows 10–10, columns 1–1). It never leaves row 10, so its queen is locked to that row.

Amber occupies 6 squares within rows 1–3, columns 8–10.

Pale yellow: 6 squares across rows 6–9, columns 4–5.

Orange covers 7 squares, spanning 3 rows and 4 columns (rows 8–10, columns 5–8).

Blue occupies 10 squares within rows 2–5, columns 6–9.

Red: 12 squares across rows 1–7, columns 4–7.

Khaki covers 17 squares, spanning 6 rows and 4 columns (rows 5–10, columns 1–4).

Mint occupies 18 squares within rows 1–5, columns 1–5.

Slate: 22 squares across rows 4–10, columns 7–10.

Where to start on this board

The most constrained color is where a solve should begin, and on #138 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 3 to column 3 — 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 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. Sea green is no more than two rows tall, so its queen is 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 #138

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3, 4, 6, 7, 8, 9 and 10 are the narrowest, touched by only 4 colors. Columns 2 and 10 are the tightest columns, with 2 colors.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 4 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

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.

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.

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 #138 is rated hard

The rating reflects how much the board concedes up front. Puzzle #138 has four regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 4 colors, and the gap between its largest and smallest regions is 21 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.

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 #138 the place to look for one is around column 2.

As a sanity check rather than a spoiler: the finished solution to #138 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.