10x10 Queens Puzzle #187

About 10x10 Queens puzzle #187

10x10 puzzle #187 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, #187 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 #187 are: mint (1 square), khaki (2 squares), blue (5 squares), amber (7 squares), red (8 squares), sea green (9 squares), slate (10 squares), pale yellow (12 squares), indigo (18 squares), orange (28 squares). An even split would hand every color 10 squares, so the mint region is running 9 squares under that average while the orange region is the roomiest area on the grid with 28 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Mint has only 1 place to put its queen; orange has 28. The spread between them on this board is 27 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. Mint is the tidiest shape on #187, packing 100% of its 1×1 bounding box, whereas red straggles across its 5×3 box at just 53%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

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.

Mint covers 1 square, spanning 1 row and 1 column (rows 4–4, columns 8–8). It never leaves row 4, so its queen is locked to that row.

Khaki occupies 2 squares within rows 7–8, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Blue: 5 squares across rows 3–6, columns 6–7.

Amber covers 7 squares, spanning 3 rows and 4 columns (rows 1–3, columns 7–10).

Red occupies 8 squares within rows 2–6, columns 1–3.

Sea green: 9 squares across rows 7–10, columns 1–3.

Slate covers 10 squares, spanning 5 rows and 3 columns (rows 2–6, columns 8–10).

Pale yellow occupies 12 squares within rows 4–7, columns 2–6.

Indigo: 18 squares across rows 1–4, columns 1–7.

Orange covers 28 squares, spanning 5 rows and 8 columns (rows 6–10, columns 3–10).

Where to start on this board

Open on the tightest color. Here that is mint, confined to 1 square between rows 4 and 4 and columns 8 and 8. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #187.

Because mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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. Khaki 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 #187

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 9 and 10 are the narrowest, touched by only 2 colors. Columns 4, 5, 9 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 4, meeting 6 colors, is the opposite case — the busiest line on #187 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 #187 is rated hard

Ratings track how generous the opening is. On #187 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 27-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.

The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a hard board from the rest.

If #187 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 #187 the place to look for one is around row 1.

Here is a shape check that gives nothing away: when #187 is solved, 4 queens sit on the outer edge of the grid and 6 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.