10x10 Queens Puzzle #194

About 10x10 Queens puzzle #194

Board #194 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.

Like every board on this site, #194 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 #194 are: orange (1 square), indigo (2 squares), slate (2 squares), sea green (4 squares), mint (7 squares), blue (10 squares), amber (18 squares), khaki (18 squares), red (19 squares), pale yellow (19 squares). An even split would hand every color 10 squares, so the orange region is running 9 squares under that average while the pale yellow region is the roomiest area on the grid with 19 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Orange has only 1 place to put its queen; pale yellow has 19. The spread between them on this board is 18 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 indigo region is the most compact color here, filling 100% of the 1×2 box that encloses it, while pale yellow is the most scattered at 54% of its 7×5 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

Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a medium board.

Orange occupies 1 square within rows 9–9, columns 8–8. It never leaves row 9, so its queen is locked to that row.

Indigo: 2 squares across rows 1–1, columns 2–3. It never leaves row 1, so its queen is locked to that row.

Slate covers 2 squares, spanning 2 rows and 1 column (rows 4–5, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Sea green occupies 4 squares within rows 9–10, columns 3–5.

Mint: 7 squares across rows 1–6, columns 9–10.

Blue covers 10 squares, spanning 4 rows and 4 columns (rows 1–4, columns 1–4).

Amber occupies 18 squares within rows 1–5, columns 5–9.

Khaki: 18 squares across rows 8–10, columns 1–8.

Red covers 19 squares, spanning 5 rows and 6 columns (rows 3–7, columns 1–6).

Pale yellow occupies 19 squares within rows 4–10, columns 6–10.

Where to start on this board

The most constrained color is where a solve should begin, and on #194 that is orange with 1 square. It reaches from row 9 to row 9 and from column 8 to column 8 — 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 orange never leaves row 9, its queen must be in that row — and that instantly settles row 9 for everyone else. Sweep along it and X out every square that is not orange. 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. Indigo, slate and sea green 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 #194

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 7 and 8 are the narrowest, touched by only 2 colors. Column 10 is the tightest column, with 2 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 #194 and the last one you should expect to crack.

Working through the middle game

Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.

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 #194 is rated medium

The medium label comes out of four measurable things about #194: four of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 18. Boards that score tight on those measures give up their first queen fast.

Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.

If #194 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 #194 the place to look for one is around row 7.

Here is a shape check that gives nothing away: when #194 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.