9x9 Queens Puzzle #36
Difficulty: hard. 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 23-step reasoning is written out below.
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Step-by-step solution for puzzle #36
What follows is a complete solve of puzzle #36, worked the way a person works it rather than the way a computer does. It runs to 23 deductions — 9 queen placements and 14 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: region confined to one line, pairing argument on rows, last square in a region, region shadow, proof by contradiction.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Amber cannot leave column 7 — every one of its open squares (3 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 8 squares in all.
Step 2. All of the blue region's remaining squares (2 open squares) lie in row 3. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 10 squares come off.
Step 3. Amber cannot leave column 7 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 2 squares in all.
Step 4. All of the mint region's remaining squares (3 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.
Step 5. Red cannot leave column 1 — every one of its open squares (3 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 6 squares in all.
The board after step 5 — 0 of 9 queens placed. Step 6. Here is the pairing argument. The indigo, amber, and mint regions have no open squares outside rows 1, 2, and 4 — 3 colours confined to 3 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. Slate cannot leave row 5 — every one of its open squares (3 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 3 squares in all.
Step 8. The red region is down to one open square, at row 6, column 1. Every colour holds exactly one queen, so that square is forced. Row 6 and column 1 are now spoken for, along with the squares touching that queen.
Step 9. The pale yellow region has 4 squares left, spread across rows 7 and 8, columns 3, 5, 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.
Step 10. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 2. Follow the forced moves that would create and the orange region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 11. Test row 1, column 3 for indigo: place a queen there, run the forced consequences through, and the pale yellow region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 11 — 1 of 9 queens placed. Step 12. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 5. Follow the forced moves that would create and the blue region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 13. Test row 2, column 2 for indigo: place a queen there, run the forced consequences through, and column 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 14. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 3. Follow the forced moves that would create and column 8 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 15. Test row 4, column 2 for indigo: place a queen there, run the forced consequences through, and column 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 16. 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: 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 7; blue has one square left, so its queen takes row 3, column 5; mint has one square left, so its queen takes row 4, column 9; slate has one square left, so its queen takes row 5, column 6; pale yellow has one square left, so its queen takes row 7, column 3; orange has one square left, so its queen takes row 9, column 2; khaki has one square left, so its queen takes row 8, column 8. 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 8, column 8.
The board after step 16 — 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 #36
This is puzzle number 36 in the 9x9 collection, rated hard. The grid holds 81 squares split between nine colored regions, and a finished board carries nine queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
As with every puzzle in this collection, #36 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 #36 are: amber (3 squares), blue (3 squares), red (3 squares), mint (7 squares), orange (11 squares), pale yellow (12 squares), indigo (13 squares), slate (14 squares), khaki (15 squares). An even split would hand every color 9 squares, so the amber region is running 6 squares under that average while the khaki region is the roomiest area on the grid with 15 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 3 squares while khaki has 15 to spare, a gap of 12. Attack the cramped end and the roomy regions tend to resolve themselves.
Square count is only half the story — how a color is arranged counts too. Amber is the tidiest shape on #36, packing 100% of its 3×1 bounding box, whereas slate straggles across its 5×7 box at just 40%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a hard board.
Amber: 3 squares across rows 1–3, columns 7–7. It sits wholly inside column 7, so its queen must be in that column.
Blue covers 3 squares, spanning 2 rows and 2 columns (rows 2–3, columns 5–6).
Red occupies 3 squares within rows 4–6, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Mint: 7 squares across rows 1–4, columns 8–9.
Orange covers 11 squares, spanning 4 rows and 4 columns (rows 6–9, columns 1–4).
Pale yellow occupies 12 squares within rows 4–8, columns 2–6.
Indigo: 13 squares across rows 1–4, columns 1–6.
Slate covers 14 squares, spanning 5 rows and 7 columns (rows 2–6, columns 3–9).
Khaki occupies 15 squares within rows 6–9, columns 4–9.
Where to start on this board
Begin where the choices are fewest. On this board that means amber, which owns just 3 squares inside a 3×1 window running from row 1 to row 3 and column 7 to column 7. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because amber stays inside column 7, its queen belongs to that column and no other region may use it. Cross out the rest of column 7 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
Work outward to the other flat regions next. Blue 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 #36
A quick way to find the next move is to count how many different colors touch each line. On this board, row 9 is the narrowest, touched by only 2 colors. Columns 1, 2, 7, 8 and 9 are the tightest columns, with 3 colors.
A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 8 rows. Row 6 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.
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.
Resist the plausible-looking queen. On #36 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #36 is rated hard
The hard label comes out of four measurable things about #36: two of its regions are locked to one row or column, its smallest color spans 3 squares, its least crowded row meets 2 colors, and its largest and smallest regions differ by 12. Boards that score tight on those measures give up their first queen fast.
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 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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 9 on this board.
A non-spoiling distribution check: solved, #36 carries 4 queens on the perimeter against 5 queens inside, with the corners empty. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.
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 #36?
Puzzle #36 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #36 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #36, 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 #36?
Start with the amber region — the most constrained color on this board, with 3 squares spanning rows 1–3 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.