8x8 Queens Puzzle #30
Difficulty: medium. Place 8 queens on this 8x8 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 16-step reasoning is written out below.
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Step-by-step solution for puzzle #30
What follows is a complete solve of puzzle #30, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 8 queen placements and 8 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 proof by contradiction.
Techniques this board needs: last square in a region, region shadow, region confined to one line, pairing argument on rows, proof by contradiction.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Only row 5, column 8 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 8, the rest of the slate region and every square adjacent to it — diagonals included.
Step 2. The khaki region is down to one open square, at row 8, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 1, the remaining khaki squares, and the ring of neighbours around row 8, column 1.
Step 3. Only row 6, column 2 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 2, in red and in the ring around that square is now dead.
Step 4. Wherever amber ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 4 — 3 of 8 queens placed. Step 5. Blue cannot leave column 3 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 4 squares in all.
Step 6. The indigo region has 3 squares left, spread across rows 1 and 2, columns 4 and 5. 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.
Step 7. Wherever mint ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 8. Here is the pairing argument. The blue and mint regions have no open squares outside rows 3 and 4 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 1 square come off in one stroke.
The board after step 8 — 3 of 8 queens placed. Step 9. The amber region has 3 squares left, spread across rows 1 and 2, columns 6 and 7. 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.
Step 10. All of the indigo region's remaining squares (2 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
Step 11. Test row 2, column 4 for indigo: place a queen there, run the forced consequences through, and row 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 12. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: indigo has one square left, so its queen takes row 1, column 4; amber has one square left, so its queen takes row 2, column 6; mint has one square left, so its queen takes row 4, column 5; blue has one square left, so its queen takes row 3, column 3; pale yellow has one square left, so its queen takes row 7, column 7. 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 7.
The board after step 12 — 8 of 8 queens placed.
That completes the board: 8 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 8x8 Queens puzzle #30
Board #30 of the 8x8 bank carries a medium rating. Its 64 squares are shared between eight colored regions, and the solution places eight queens: one per row, one per column, one per region, none of them side by side or corner to corner.
As with every puzzle in this collection, #30 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 #30 are: slate (1 square), khaki (1 square), amber (5 squares), red (6 squares), indigo (8 squares), blue (11 squares), pale yellow (13 squares), mint (19 squares). An even split would hand every color 8 squares, so the slate region is running 7 squares under that average while the mint region is the roomiest area on the grid with 19 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 mint with 19. A 18-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 slate is the most solid, filling 100% of its 1×1 box, and mint the most diffuse at 45% of its 7×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
Slate: 1 square across rows 5–5, columns 8–8. It never leaves row 5, so its queen is locked to that row.
Khaki covers 1 square, spanning 1 row and 1 column (rows 8–8, columns 1–1). It never leaves row 8, so its queen is locked to that row.
Amber occupies 5 squares within rows 1–3, columns 6–7.
Red: 6 squares across rows 6–8, columns 1–3.
Indigo covers 8 squares, spanning 3 rows and 5 columns (rows 1–3, columns 1–5).
Blue occupies 11 squares within rows 2–5, columns 1–3.
Pale yellow: 13 squares across rows 6–8, columns 4–8.
Mint covers 19 squares, spanning 7 rows and 6 columns (rows 1–7, columns 3–8).
Where to start on this board
Slate is the natural opening on #30: it is the smallest region on the grid at 1 square, boxed into rows 5–5 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 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. Khaki is no more than two rows tall, so its queen is 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 #30
A quick way to find the next move is to count how many different colors touch each line. On this board, row 4 is the narrowest, touched by only 2 colors. Columns 2, 4, 5, 6, 7 and 8 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 7 rows. Row 3, meeting 4 colors, is the opposite case — the busiest line on #30 and the last one you should expect to crack.
Working through the middle game
The middle of #30 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 eight colors looking for one with a single square left; sweep the eight rows and eight columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 8x8 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 #30 is rated medium
Ratings track how generous the opening is. On #30 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 18-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.
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 8x8
At 8x8 the whole grid stays within a single glance, which makes systematic scanning practical. Read the rows in order and count the colors in each — a row touched by two colors is far more constrained than one touched by five, and that is usually where the next forced move hides. The board is small enough that re-scanning from the top after each placement costs almost nothing.
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 #30, start hunting for it near row 4.
Here is a shape check that gives nothing away: when #30 is solved, 3 queens sit on the outer edge of the grid and 5 sit inside it, with 1 queen 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
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- 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.
- 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 8x8 Queens puzzle #30?
Puzzle #30 is rated medium. 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 8x8 puzzle #30 have only one solution?
Yes. Every puzzle in the 8x8 collection, including #30, has been verified to have exactly one valid arrangement of eight queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #30?
Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.