11x11 Queens Puzzle #37

About 11x11 Queens puzzle #37

Puzzle #37 is one of the easy-rated boards in the 11x11 set. Eleven colors divide its 121 squares, and you win by placing eleven queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.

There is exactly one way to finish #37, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these eleven regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #37 are: indigo (1 square), red (1 square), teal (1 square), slate (3 squares), blue (7 squares), khaki (8 squares), mint (11 squares), orange (14 squares), pale yellow (21 squares), sea green (21 squares), amber (33 squares). An even split would hand every color 11 squares, so the indigo region is running 10 squares under that average while the amber 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 indigo, with 1 square, is under far more pressure than amber with 33. A 32-square gap like this one rewards players who start where the choices are fewest.

Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×1 box that encloses it, while orange is the most scattered at 40% of its 5×7 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

Below is every color on #37 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

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

Red occupies 1 square within rows 6–6, columns 7–7. It never leaves row 6, so its queen is locked to that row.

Teal: 1 square across rows 11–11, columns 6–6. It never leaves row 11, so its queen is locked to that row.

Slate covers 3 squares, spanning 3 rows and 1 column (rows 5–7, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Blue occupies 7 squares within rows 1–5, columns 10–11.

Khaki: 8 squares across rows 7–11, columns 1–2.

Mint covers 11 squares, spanning 4 rows and 3 columns (rows 1–4, columns 1–3).

Orange occupies 14 squares within rows 7–11, columns 2–8.

Pale yellow: 21 squares across rows 3–10, columns 2–6.

Sea green covers 21 squares, spanning 5 rows and 5 columns (rows 7–11, columns 7–11).

Amber occupies 33 squares within rows 1–6, columns 3–11.

Where to start on this board

Indigo is the natural opening on #37: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 10–10. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.

Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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 teal 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 #37

A quick way to find the next move is to count how many different colors touch each line. On this board, row 2 is the narrowest, touched by only 3 colors. Column 9 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 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 10 rows. By contrast row 7 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

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.

The whole middle game reduces to two sweeps. Sweep the eleven colors looking for one with a single square left; sweep the eleven rows and eleven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 11x11 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 #37 is rated easy

What earns a board its rating is the shape of its opening. Here that opening offers seven single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 32. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

Stuck on #37 despite the easy rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.

Techniques that work well at 11x11

At 11x11 you are managing 121 squares, and holding the whole board in your head will not work. Split it into quadrants and solve locally. Most useful deductions on a grid this large are regional — a color pinched between two placed queens, or a band of rows that two regions must share. Trust the marks on the board over your memory.

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 #37, start hunting for it near column 9.

As a sanity check rather than a spoiler: the finished solution to #37 puts 4 queens on the outer border and 7 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.