10x10 Queens Puzzle #294

About 10x10 Queens puzzle #294

Board #294 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.

As with every puzzle in this collection, #294 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 #294 are: blue (1 square), orange (3 squares), sea green (3 squares), indigo (4 squares), pale yellow (6 squares), slate (7 squares), amber (11 squares), khaki (11 squares), mint (27 squares), red (27 squares). An even split would hand every color 10 squares, so the blue region is running 9 squares under that average while the red region is the roomiest area on the grid with 27 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 blue, with 1 square, is under far more pressure than red with 27. A 26-square gap like this one rewards players who start where the choices are fewest.

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 khaki is the most scattered at 55% of its 4×5 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 easy board.

Blue: 1 square across rows 3–3, columns 7–7. It never leaves row 3, so its queen is locked to that row.

Orange covers 3 squares, spanning 2 rows and 2 columns (rows 9–10, columns 8–9).

Sea green occupies 3 squares within rows 10–10, columns 4–6. It never leaves row 10, so its queen is locked to that row.

Indigo: 4 squares across rows 1–2, columns 1–3.

Pale yellow covers 6 squares, spanning 4 rows and 2 columns (rows 7–10, columns 9–10).

Slate occupies 7 squares within rows 2–6, columns 1–2.

Amber: 11 squares across rows 1–3, columns 7–10.

Khaki covers 11 squares, spanning 4 rows and 5 columns (rows 7–10, columns 1–5).

Mint occupies 27 squares within rows 1–8, columns 2–7.

Red: 27 squares across rows 4–10, columns 4–10.

Where to start on this board

The most constrained color is where a solve should begin, and on #294 that is blue with 1 square. It reaches from row 3 to row 3 and from column 7 to column 7 — 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. Sea green, indigo 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 #294

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 4, 5 and 6 are the narrowest, touched by only 3 colors. Columns 1, 3, 6, 8 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 3 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 10, meeting 5 colors, is the opposite case — the busiest line on #294 and the last one you should expect to crack.

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.

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 #294 is rated easy

The easy label comes out of four measurable things about #294: three of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 26. 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.

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.

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 #294, start hunting for it near row 1.

Here is a shape check that gives nothing away: when #294 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.