9x9 Queens Puzzle #21

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 #21

What follows is a complete solve of puzzle #21, 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 columns.

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

Show the full walkthrough (7 steps) — contains the solution

  1. Step 1. Only row 3, column 7 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 7, the rest of the blue region and every square adjacent to it — diagonals included.

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

  3. Step 3. Red cannot leave row 6 — every one of its open squares (2 open squares) sits there. That row belongs to red, so every square on it owned by another colour is dead: 5 squares in all.

  4. Step 4. Wherever khaki ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

    Puzzle #21 after step 4: 1 of 9 queens placed
    The board after step 4 — 1 of 9 queens placed.

  5. Step 5. Indigo, mint, and red are now boxed into columns 6, 8, and 9 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 10 squares at once.

  6. Step 6. All of the orange region's remaining squares (2 open squares) lie in row 9. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.

  7. Step 7. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: row 8 is down to one square, putting a queen on row 8, column 2; slate has one square left, so its queen takes row 5, column 1; mint has one square left, so its queen takes row 4, column 9; indigo has one square left, so its queen takes row 1, column 8; red has one square left, so its queen takes row 6, column 6; pale yellow has one square left, so its queen takes row 7, column 4; orange has one square left, so its queen takes row 9, column 5; amber has one square left, so its queen takes row 2, column 3. 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 2, column 3.

    Puzzle #21 after step 7: 9 of 9 queens placed
    The board after step 7 — 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 #21

Puzzle #21 is one of the easy-rated boards in the 9x9 set. Nine colors divide its 81 squares, and you win by placing nine 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 #21, 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 #21 are: blue (1 square), slate (3 squares), red (3 squares), indigo (4 squares), khaki (6 squares), mint (12 squares), pale yellow (15 squares), orange (15 squares), amber (22 squares). An even split would hand every color 9 squares, so the blue region is running 8 squares under that average while the amber region is the roomiest area on the grid with 22 squares.

Every color takes one queen and one only, whatever its size — which is why the 21 square difference between blue and amber 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.

Look at arrangement as well as area. Blue is the densest region on this grid at 100% of a 1×1 frame; orange is the loosest, occupying only 63% of its 4×6 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

Region by region

The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.

Blue: 1 square across rows 3–3, columns 7–7. It never leaves row 3, 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.

Red occupies 3 squares within rows 6–6, columns 6–8. It never leaves row 6, so its queen is locked to that row.

Indigo: 4 squares across rows 1–2, columns 7–9.

Khaki covers 6 squares, spanning 3 rows and 3 columns (rows 7–9, columns 1–3).

Mint occupies 12 squares within rows 3–5, columns 5–9.

Pale yellow: 15 squares across rows 4–9, columns 2–5.

Orange covers 15 squares, spanning 4 rows and 6 columns (rows 6–9, columns 4–9).

Amber occupies 22 squares within rows 1–4, columns 1–8.

Where to start on this board

The most constrained color is where a solve should begin, and on #21 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. Red and indigo 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 #21

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 2 are the narrowest, touched by only 2 colors. Columns 1, 2, 3, 4 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 4 colors, is the opposite case — the busiest line on #21 and the last one you should expect to crack.

Working through the middle game

The middle of #21 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 nine colors looking for one with a single square left; sweep the nine rows and nine columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 9x9 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 #21 is rated easy

Ratings track how generous the opening is. On #21 you get four regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 21-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.

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 #21 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.

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 #21 is row 1.

Here is a shape check that gives nothing away: when #21 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 #21?

Puzzle #21 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.

Does 9x9 puzzle #21 have only one solution?

Yes. Every puzzle in the 9x9 collection, including #21, 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 #21?

Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.