9x9 Queens Puzzle #264
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 17-step reasoning is written out below.
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Step-by-step solution for puzzle #264
What follows is a complete solve of puzzle #264, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 9 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 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 (9 steps) — contains the solution
Step 1. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 9 squares in all.
Step 2. All of the amber region's remaining squares (2 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 7 squares come off.
Step 3. The orange region has 4 squares left, spread across rows 8 and 9, columns 3–5. 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.
The board after step 3 — 0 of 9 queens placed. Step 4. Wherever khaki 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 5. Khaki and orange are now boxed into rows 8 and 9 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 8 squares at once.
Step 6. Here is the pairing argument. The indigo and khaki regions have no open squares outside columns 1 and 2 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 10 squares come off in one stroke.
The board after step 6 — 0 of 9 queens placed. Step 7. Slate cannot leave column 3 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 5 squares in all.
Step 8. All of the blue 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: 6 squares come off.
Step 9. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: orange has one square left, so its queen takes row 9, column 5; khaki has one square left, so its queen takes row 8, column 2; indigo has one square left, so its queen takes row 1, column 1; slate has one square left, so its queen takes row 5, column 3; blue has one square left, so its queen takes row 3, column 4; mint has one square left, so its queen takes row 4, column 6; red has one square left, so its queen takes row 6, column 7; pale yellow has one square left, so its queen takes row 7, column 9; amber has one square left, so its queen takes row 2, 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 2, column 8.
The board after step 9 — 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 #264
This is puzzle number 264 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.
There is exactly one way to finish #264, 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 #264 are: indigo (2 squares), khaki (4 squares), orange (4 squares), amber (6 squares), red (8 squares), slate (10 squares), mint (12 squares), blue (15 squares), pale yellow (20 squares). An even split would hand every color 9 squares, so the indigo region is running 7 squares under that average while the pale yellow region is the roomiest area on the grid with 20 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 2 squares while pale yellow has 20 to spare, a gap of 18. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×2 frame; red is the loosest, occupying only 44% of its 3×6 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.
Indigo: 2 squares across rows 1–1, columns 1–2. It never leaves row 1, so its queen is locked to that row.
Khaki covers 4 squares, spanning 2 rows and 3 columns (rows 8–9, columns 1–3).
Orange occupies 4 squares within rows 8–9, columns 3–5.
Amber: 6 squares across rows 1–2, columns 6–9.
Red covers 8 squares, spanning 3 rows and 6 columns (rows 4–6, columns 3–8).
Slate occupies 10 squares within rows 4–8, columns 1–3.
Mint: 12 squares across rows 2–5, columns 4–9.
Blue covers 15 squares, spanning 4 rows and 5 columns (rows 1–4, columns 1–5).
Pale yellow occupies 20 squares within rows 4–9, columns 4–9.
Where to start on this board
Begin where the choices are fewest. On this board that means indigo, which owns just 2 squares inside a 1×2 window running from row 1 to row 1 and column 1 to column 2. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Amber, 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 #264
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 7 are the narrowest, touched by only 2 colors. Column 9 is the tightest column, 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 4 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
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
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 #264 is rated hard
Ratings track how generous the opening is. On #264 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 18-square gap between the largest and smallest regions. Those four numbers largely explain the hard 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 hard board from the rest.
Stuck on #264 despite the hard 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.
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 3 on this board.
As a sanity check rather than a spoiler: the finished solution to #264 puts 3 queens on the outer border and 6 queens in the interior, 1 of them in an actual 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 #264?
Puzzle #264 is rated hard. 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 #264 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #264, 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 #264?
Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–1 and columns 1–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.