9x9 Queens Puzzle #39

Difficulty: easy. Place 9 queens on this 9x9 board so that no two share a row, column or color region and none touch — including diagonally. This puzzle has exactly one solution, and it has been verified solvable by deduction alone: the complete 14-step reasoning is written out below.

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Step-by-step solution for puzzle #39

What follows is a complete solve of puzzle #39, worked the way a person works it rather than the way a computer does. It runs to 14 deductions — 9 queen placements and 5 elimination steps that narrow the board between them. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the pairing argument on rows.

Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows.

Show the full walkthrough (8 steps) — contains the solution

  1. Step 1. Only row 9, column 8 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 8, the rest of the orange region and every square adjacent to it — diagonals included.

  2. Step 2. All of the indigo region's remaining squares (3 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.

  3. Step 3. Only row 2, column 9 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 9, in amber and in the ring around that square is now dead.

  4. Step 4. All of the red region's remaining squares (2 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.

    Puzzle #39 after step 4: 2 of 9 queens placed
    The board after step 4 — 2 of 9 queens placed.

  5. Step 5. Pale yellow cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 6 squares in all.

  6. Step 6. The khaki region has 3 squares left, spread across rows 7 and 8, columns 3 and 6. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.

  7. Step 7. Pale yellow and khaki are now boxed into rows 7 and 8 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 4 squares at once.

  8. Step 8. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: red has one square left, so its queen takes row 6, column 7; slate has one square left, so its queen takes row 5, column 2; blue has one square left, so its queen takes row 3, column 3; khaki has one square left, so its queen takes row 8, column 6; mint has one square left, so its queen takes row 4, column 5; indigo has one square left, so its queen takes row 1, column 4; pale yellow has one square left, so its queen takes row 7, column 1. After each one, clear that row, that column, the rest of the colour and the ring of neighbours before looking for the next. The last of them seats the queen on row 7, column 1.

    Puzzle #39 after step 8: 9 of 9 queens placed
    The board after step 8 — 9 of 9 queens placed.

That completes the board: 9 queens seated, one in every row, one in every column and one in every colour, with no two of them touching. Notice that nothing in the sequence above required a guess — each placement followed from an elimination already on the grid, which is what “single solution, solvable by logic” actually means in practice.

About 9x9 Queens puzzle #39

9x9 puzzle #39 is rated easy. Nine colored regions cover its 81 squares, and your job is to seat nine queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

There is exactly one way to finish #39, 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 nine regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #39 are: orange (1 square), indigo (3 squares), pale yellow (4 squares), amber (5 squares), slate (6 squares), red (10 squares), blue (12 squares), khaki (12 squares), mint (28 squares). An even split would hand every color 9 squares, so the orange region is running 8 squares under that average while the mint region is the roomiest area on the grid with 28 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; mint has 28. The spread between them on this board is 27 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 1×3 box that encloses it, while slate is the most scattered at 50% of its 3×4 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 #39 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.

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

Indigo covers 3 squares, spanning 1 row and 3 columns (rows 1–1, columns 4–6). It never leaves row 1, so its queen is locked to that row.

Pale yellow occupies 4 squares within rows 7–9, columns 1–2.

Amber: 5 squares across rows 1–2, columns 7–9.

Slate covers 6 squares, spanning 3 rows and 4 columns (rows 5–7, columns 1–4).

Red occupies 10 squares within rows 4–9, columns 7–9.

Blue: 12 squares across rows 1–5, columns 1–3.

Khaki covers 12 squares, spanning 3 rows and 7 columns (rows 7–9, columns 2–8).

Mint occupies 28 squares within rows 2–8, columns 3–9.

Where to start on this board

Orange is the natural opening on #39: it is the smallest region on the grid at 1 square, boxed into rows 9–9 and columns 8–8. 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 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 and amber 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 #39

A quick way to find the next move is to count how many different colors touch each line. On this board, row 3 is the narrowest, touched by only 2 colors. Columns 1, 5, 6 and 9 are the tightest columns, with 3 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 8 rows. Row 7, meeting 5 colors, is the opposite case — the busiest line on #39 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.

With nine queens to seat and nine regions competing over 81 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 9x9 boards quickly and players who stall halfway.

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 #39 is rated easy

What earns a board its rating is the shape of its opening. Here that opening offers three single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 27. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a easy board from the rest.

Stuck on #39 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 9x9

At 9x9 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 #39 the place to look for one is around row 3.

Here is a shape check that gives nothing away: when #39 is solved, 4 queens sit on the outer edge of the grid and 5 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.

How to play Queens

  1. The board is an N×N grid divided into N colored regions.
  2. Place exactly one queen (♛) in each row, each column, and each colored region.
  3. No two queens may touch each other — not horizontally, vertically, or diagonally.
  4. Tap once to mark a square with an X (where a queen can't go), tap again to place a queen, and once more to clear it.
  5. There is always exactly one valid solution. Use logic, never guessing, to find it.

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Questions about this puzzle

How hard is 9x9 Queens puzzle #39?

Puzzle #39 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.

Does 9x9 puzzle #39 have only one solution?

Yes. Every puzzle in the 9x9 collection, including #39, has been verified to have exactly one valid arrangement of nine queens, and it can always be reached by logic alone — no guessing or backtracking needed.

Where should I place my first queen on puzzle #39?

Start with the orange region — the most constrained color on this board, with 1 square spanning rows 9–9 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.