10x10 Queens Puzzle #86

About 10x10 Queens puzzle #86

Board #86 of the 10x10 bank carries a medium 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, #86 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 #86 are: blue (1 square), red (1 square), orange (3 squares), slate (4 squares), khaki (7 squares), sea green (7 squares), indigo (10 squares), amber (13 squares), mint (16 squares), pale yellow (38 squares). An even split would hand every color 10 squares, so the blue region is running 9 squares under that average while the pale yellow region is the roomiest area on the grid with 38 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Blue has only 1 place to put its queen; pale yellow has 38. The spread between them on this board is 37 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Shape matters as much as size. The blue region is the most compact color here, filling 100% of the 1×1 box that encloses it, while mint is the most scattered at 44% of its 6×6 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

Here is the full footprint of every color on puzzle #86, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Blue occupies 1 square within rows 3–3, columns 5–5. It never leaves row 3, so its queen is locked to that row.

Red: 1 square across rows 6–6, columns 2–2. It never leaves row 6, so its queen is locked to that row.

Orange covers 3 squares, spanning 2 rows and 2 columns (rows 9–10, columns 2–3).

Slate occupies 4 squares within rows 3–6, columns 10–10. It sits wholly inside column 10, so its queen must be in that column.

Khaki: 7 squares across rows 7–9, columns 7–10.

Sea green covers 7 squares, spanning 3 rows and 3 columns (rows 8–10, columns 8–10).

Indigo occupies 10 squares within rows 1–3, columns 4–8.

Amber: 13 squares across rows 1–5, columns 1–4.

Mint covers 16 squares, spanning 6 rows and 6 columns (rows 1–6, columns 5–10).

Pale yellow occupies 38 squares within rows 3–10, columns 1–8.

Where to start on this board

The most constrained color is where a solve should begin, and on #86 that is blue with 1 square. It reaches from row 3 to row 3 and from column 5 to column 5 — 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 blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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. 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 #86

A quick way to find the next move is to count how many different colors touch each line. On this board, row 7 is the narrowest, touched by only 2 colors. Column 1 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 3 meets 6 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 #86 is rated medium

The rating reflects how much the board concedes up front. Puzzle #86 has five regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 37 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 #86 the place to look for one is around row 7.

As a sanity check rather than a spoiler: the finished solution to #86 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.