10x10 Queens Puzzle #38

About 10x10 Queens puzzle #38

Board #38 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.

Like every board on this site, #38 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 #38 are: pale yellow (1 square), slate (2 squares), orange (2 squares), sea green (6 squares), blue (8 squares), amber (10 squares), indigo (11 squares), khaki (12 squares), red (22 squares), mint (26 squares). An even split would hand every color 10 squares, so the pale yellow region is running 9 squares under that average while the mint region is the roomiest area on the grid with 26 squares.

Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So pale yellow, with 1 square, is under far more pressure than mint with 26. A 25-square gap like this one rewards players who start where the choices are fewest.

Shape matters as much as size. The slate region is the most compact color here, filling 100% of the 2×1 box that encloses it, while mint is the most scattered at 54% of its 6×8 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.

Region by region

Below is every color on #38 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

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.

Slate: 2 squares across rows 5–6, columns 10–10. It sits wholly inside column 10, so its queen must be in that column.

Orange covers 2 squares, spanning 2 rows and 1 column (rows 9–10, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Sea green occupies 6 squares within rows 7–10, columns 9–10.

Blue: 8 squares across rows 1–3, columns 1–3.

Amber covers 10 squares, spanning 3 rows and 6 columns (rows 1–3, columns 3–8).

Indigo occupies 11 squares within rows 1–5, columns 7–10.

Khaki: 12 squares across rows 6–10, columns 6–9.

Red covers 22 squares, spanning 5 rows and 7 columns (rows 6–10, columns 1–7).

Mint occupies 26 squares within rows 2–7, columns 1–8.

Where to start on this board

The most constrained color is where a solve should begin, and on #38 that is pale yellow with 1 square. It reaches from row 7 to row 7 and from column 6 to column 6 — 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 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. Slate 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 #38

A quick way to find the next move is to count how many different colors touch each line. On this board, row 4 is the narrowest, touched by only 2 colors. Columns 2, 4, 5, 9 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 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.

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 #38 is rated easy

What earns a board its rating is the shape of its opening. Here that opening offers four single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 25. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

If #38 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.

Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #38, start hunting for it near row 4.

As a sanity check rather than a spoiler: the finished solution to #38 puts 4 queens on the outer border and 6 queens in the interior, with none in a 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.