10x10 Queens Puzzle #131

About 10x10 Queens puzzle #131

10x10 puzzle #131 is rated hard. Ten colored regions cover its 100 squares, and your job is to seat ten queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

Like every board on this site, #131 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.

How the colors divide this board

From smallest to largest, the regions on puzzle #131 are: pale yellow (1 square), sea green (1 square), indigo (5 squares), red (7 squares), amber (8 squares), blue (8 squares), slate (10 squares), mint (12 squares), khaki (13 squares), orange (35 squares). An even split would hand every color 10 squares, so the pale yellow region is running 9 squares under that average while the orange region is the roomiest area on the grid with 35 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Pale yellow has only 1 place to put its queen; orange has 35. The spread between them on this board is 34 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Look at arrangement as well as area. Pale yellow is the densest region on this grid at 100% of a 1×1 frame; blue is the loosest, occupying only 40% of its 4×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

Region by region

The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.

Pale yellow occupies 1 square within rows 7–7, columns 6–6. It never leaves row 7, so its queen is locked to that row.

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

Indigo covers 5 squares, spanning 2 rows and 4 columns (rows 1–2, columns 6–9).

Red occupies 7 squares within rows 3–7, columns 1–2.

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

Blue covers 8 squares, spanning 4 rows and 5 columns (rows 1–4, columns 3–7).

Slate occupies 10 squares within rows 4–6, columns 2–7.

Mint: 12 squares across rows 1–4, columns 7–10.

Khaki covers 13 squares, spanning 5 rows and 3 columns (rows 5–9, columns 8–10).

Orange occupies 35 squares within rows 5–10, columns 1–9.

Where to start on this board

Open on the tightest color. Here that is pale yellow, confined to 1 square between rows 7 and 7 and columns 6 and 6. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #131.

Because pale yellow never leaves row 7, its queen must be in that row — and that instantly settles row 7 for everyone else. Sweep along it and X out every square that is not pale yellow. 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 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 #131

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 8, 9 and 10 are the narrowest, touched by only 2 colors. Columns 1, 5 and 10 are the tightest columns, with 3 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 7 meets 4 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.

Keep alternating between two views of the board. In the region view you ask which of the ten colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 10x10 board unwinds.

The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.

Why puzzle #131 is rated hard

Ratings track how generous the opening is. On #131 you get four regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 34-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.

Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.

If #131 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.

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 #131 the place to look for one is around row 8.

As a sanity check rather than a spoiler: the finished solution to #131 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.