10x10 Queens Puzzle #57

About 10x10 Queens puzzle #57

Puzzle #57 is one of the easy-rated boards in the 10x10 set. Ten colors divide its 100 squares, and you win by placing ten queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.

As with every puzzle in this collection, #57 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 #57 are: slate (1 square), indigo (2 squares), sea green (2 squares), khaki (3 squares), mint (7 squares), amber (8 squares), blue (9 squares), orange (13 squares), red (22 squares), pale yellow (33 squares). An even split would hand every color 10 squares, so the slate region is running 9 squares under that average while the pale yellow region is the roomiest area on the grid with 33 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 slate, with 1 square, is under far more pressure than pale yellow with 33. A 32-square gap like this one rewards players who start where the choices are fewest.

A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×2 box, and blue the most diffuse at 60% of its 5×3 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

Region by region

Here is the full footprint of every color on puzzle #57, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Slate: 1 square across rows 5–5, columns 10–10. It never leaves row 5, so its queen is locked to that row.

Indigo covers 2 squares, spanning 1 row and 2 columns (rows 1–1, columns 8–9). It never leaves row 1, so its queen is locked to that row.

Sea green occupies 2 squares within rows 10–10, columns 1–2. It never leaves row 10, so its queen is locked to that row.

Khaki: 3 squares across rows 7–9, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Mint covers 7 squares, spanning 5 rows and 2 columns (rows 1–5, columns 7–8).

Amber occupies 8 squares within rows 1–4, columns 4–6.

Blue: 9 squares across rows 1–5, columns 8–10.

Orange covers 13 squares, spanning 3 rows and 6 columns (rows 8–10, columns 2–7).

Red occupies 22 squares within rows 5–10, columns 5–10.

Pale yellow: 33 squares across rows 1–8, columns 1–6.

Where to start on this board

Begin where the choices are fewest. On this board that means slate, which owns just 1 square inside a 1×1 window running from row 5 to row 5 and column 10 to column 10. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

Because slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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 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 #57

A quick way to find the next move is to count how many different colors touch each line. On this board, row 6 is the narrowest, touched by only 2 colors. Column 3 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 5 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.

The whole middle game reduces to two sweeps. Sweep the ten colors looking for one with a single square left; sweep the ten rows and ten columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 10x10 solve from a stalled one.

A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.

Why puzzle #57 is rated easy

The rating reflects how much the board concedes up front. Puzzle #57 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 32 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.

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 #57, start hunting for it near row 6.

As a sanity check rather than a spoiler: the finished solution to #57 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.