10x10 Queens Puzzle #61

About 10x10 Queens puzzle #61

Puzzle #61 is one of the medium-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.

Like every board on this site, #61 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 #61 are: sea green (1 square), indigo (3 squares), orange (5 squares), slate (7 squares), khaki (7 squares), amber (11 squares), pale yellow (12 squares), blue (13 squares), mint (20 squares), red (21 squares). An even split would hand every color 10 squares, so the sea green region is running 9 squares under that average while the red region is the roomiest area on the grid with 21 squares.

Every color takes one queen and one only, whatever its size — which is why the 20 square difference between sea green and red is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.

A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×3 box, and mint the most diffuse at 48% of its 6×7 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

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.

Sea green covers 1 square, spanning 1 row and 1 column (rows 10–10, columns 7–7). It never leaves row 10, so its queen is locked to that row.

Indigo occupies 3 squares within rows 1–1, columns 4–6. It never leaves row 1, so its queen is locked to that row.

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

Slate covers 7 squares, spanning 5 rows and 2 columns (rows 4–8, columns 1–2).

Khaki occupies 7 squares within rows 8–10, columns 1–3.

Amber: 11 squares across rows 1–4, columns 7–10.

Pale yellow covers 12 squares, spanning 5 rows and 3 columns (rows 5–9, columns 8–10).

Blue occupies 13 squares within rows 1–4, columns 1–4.

Mint: 20 squares across rows 2–7, columns 2–8.

Red covers 21 squares, spanning 6 rows and 6 columns (rows 5–10, columns 3–8).

Where to start on this board

Begin where the choices are fewest. On this board that means sea green, which owns just 1 square inside a 1×1 window running from row 10 to row 10 and column 7 to column 7. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

Because sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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 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 #61

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2 and 3 are the narrowest, touched by only 3 colors. Columns 1, 5, 6, 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 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 9 rows. By contrast row 10 meets 4 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

Working through the middle game

The middle of #61 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.

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.

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 #61 is rated medium

The medium label comes out of four measurable things about #61: 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 20. Boards that score tight on those measures give up their first queen fast.

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.

If #61 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.

There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #61 is row 1.

As a sanity check rather than a spoiler: the finished solution to #61 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.