9x9 Queens Puzzle #30
Difficulty: medium. 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 19-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 19 deductions — 9 queen placements and 10 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: region confined to one line, region shadow, pairing argument on rows, pairing argument on columns, last square in a region.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Blue cannot leave row 3 — every one of its open squares (2 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 11 squares in all.
Step 2. Amber is down to 3 candidates in rows 1 and 2, columns 7–9, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about amber itself.
Step 3. The khaki region has 3 squares left, spread across rows 8 and 9, columns 1 and 2. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 4. Here is the pairing argument. The khaki and orange regions have no open squares outside rows 8 and 9 — 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: 10 squares come off in one stroke.
Step 5. Pale yellow is down to 3 candidates in rows 6 and 7, columns 8 and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
The board after step 5 — 0 of 9 queens placed. Step 6. Here is the pairing argument. The indigo, amber, mint, and slate regions have no open squares outside rows 1, 2, 4, and 5 — 4 colours confined to 4 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 5 squares come off in one stroke.
Step 7. Blue and pale yellow are now boxed into columns 8 and 9 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 4 squares at once.
Step 8. The amber region is down to one open square, at row 2, column 7. Every colour holds exactly one queen, so that square is forced. Row 2 and column 7 are now spoken for, along with the squares touching that queen.
Step 9. Only row 3, column 9 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 9, the rest of the blue region and every square adjacent to it — diagonals included.
The board after step 9 — 2 of 9 queens placed. Step 10. The pale yellow region is down to one open square, at row 7, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 8, the remaining pale yellow squares, and the ring of neighbours around row 7, column 8.
Step 11. Slate is down to 4 candidates in rows 4 and 5, columns 4–6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.
Step 12. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 3 squares come off.
Step 13. Mint cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 3 squares in all.
Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: khaki has one square left, so its queen takes row 8, column 2; orange has one square left, so its queen takes row 9, column 4; indigo has one square left, so its queen takes row 1, column 5; slate has one square left, so its queen takes row 5, column 6; mint has one square left, so its queen takes row 4, column 1; red has one square left, so its queen takes row 6, 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 6, column 3.
The board after step 14 — 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 #30
Board #30 of the 9x9 bank carries a medium rating. Its 81 squares are shared between nine colored regions, and the solution places nine queens: one per row, one per column, one per region, none of them side by side or corner to corner.
Like every board on this site, #30 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 #30 are: blue (2 squares), khaki (3 squares), orange (5 squares), amber (6 squares), mint (10 squares), indigo (11 squares), slate (11 squares), pale yellow (11 squares), red (22 squares). An even split would hand every color 9 squares, so the blue region is running 7 squares under that average while the red region is the roomiest area on the grid with 22 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 blue, with 2 squares, is under far more pressure than red with 22. A 20-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 blue is the most solid, filling 100% of its 1×2 box, and red the most diffuse at 46% of its 6×8 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.
Blue: 2 squares across rows 3–3, columns 8–9. It never leaves row 3, so its queen is locked to that row.
Khaki covers 3 squares, spanning 2 rows and 2 columns (rows 8–9, columns 1–2).
Orange occupies 5 squares within rows 8–9, columns 2–4.
Amber: 6 squares across rows 1–3, columns 7–9.
Mint covers 10 squares, spanning 5 rows and 3 columns (rows 1–5, columns 1–3).
Indigo occupies 11 squares within rows 1–4, columns 3–7.
Slate: 11 squares across rows 3–5, columns 4–9.
Pale yellow covers 11 squares, spanning 4 rows and 4 columns (rows 6–9, columns 6–9).
Red occupies 22 squares within rows 4–9, columns 1–8.
Where to start on this board
Open on the tightest color. Here that is blue, confined to 2 squares between rows 3 and 3 and columns 8 and 9. Its queen has nowhere else to go, which means those 1 row and 2 columns carry more constraint than anywhere else on #30.
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. Khaki 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 #30
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 6 and 7 are the narrowest, touched by only 2 colors. Columns 1 and 5 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 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 8 rows. By contrast row 3 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.
Keep alternating between two views of the board. In the region view you ask which of the nine colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 9x9 board unwinds.
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 #30 is rated medium
Ratings track how generous the opening is. On #30 you get one region pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 2 colors, and a 20-square gap between the largest and smallest regions. Those four numbers largely explain the medium 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 medium board from the rest.
If #30 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 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.
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 6.
As a sanity check rather than a spoiler: the finished solution to #30 puts 4 queens on the outer border and 5 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.
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 9x9 Queens puzzle #30?
Puzzle #30 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 2 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #30 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #30, 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 #30?
Start with the blue region — the most constrained color on this board, with 2 squares spanning rows 3–3 and columns 8–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.