9x9 Queens Puzzle #133
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 11-step reasoning is written out below.
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Step-by-step solution for puzzle #133
What follows is a complete solve of puzzle #133, worked the way a person works it rather than the way a computer does. It runs to 11 deductions — 9 queen placements and 2 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 region confined to one line.
Techniques this board needs: last square in a region, region confined to one line, last square in a row.
Show the full walkthrough (4 steps) — contains the solution
Step 1. The board opens by cascading. Each of these placements is forced the moment the one before it is marked, so the whole run is really a single sweep repeated: indigo has one square left, so its queen takes row 1, column 2; red has one square left, so its queen takes row 6, column 1; blue has one square left, so its queen takes row 3, column 3; orange has one square left, so its queen takes row 9, column 8; amber has one square left, so its queen takes row 2, 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 2, column 7.
Step 2. All of the mint region's remaining squares (2 open squares) lie in column 6. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
The board after step 2 — 5 of 9 queens placed. Step 3. Slate cannot leave column 4 — 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: 1 square in all.
Step 4. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 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 5; slate has one square left, so its queen takes row 5, column 4; mint has one square left, so its queen takes row 4, column 6; pale yellow has one square left, so its queen takes row 7, column 9. 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 9.
The board after step 4 — 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 #133
Puzzle #133 is one of the hard-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 #133, 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 #133 are: indigo (1 square), red (1 square), orange (1 square), amber (6 squares), blue (6 squares), mint (13 squares), khaki (14 squares), pale yellow (19 squares), slate (20 squares). An even split would hand every color 9 squares, so the indigo region is running 8 squares under that average while the slate region is the roomiest area on the grid with 20 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 indigo, with 1 square, is under far more pressure than slate with 20. A 19-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; mint is the loosest, occupying only 52% of its 5×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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.
Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 2–2). It never leaves row 1, so its queen is locked to that row.
Red occupies 1 square within rows 6–6, columns 1–1. It never leaves row 6, so its queen is locked to that row.
Orange: 1 square across rows 9–9, columns 8–8. It never leaves row 9, so its queen is locked to that row.
Amber covers 6 squares, spanning 2 rows and 4 columns (rows 1–2, columns 6–9).
Blue occupies 6 squares within rows 1–3, columns 1–3.
Mint: 13 squares across rows 1–5, columns 3–7.
Khaki covers 14 squares, spanning 5 rows and 5 columns (rows 5–9, columns 2–6).
Pale yellow occupies 19 squares within rows 2–9, columns 7–9.
Slate: 20 squares across rows 3–9, columns 1–5.
Where to start on this board
The most constrained color is where a solve should begin, and on #133 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 2 to column 2 — 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 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. Red, orange and amber 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 #133
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 4, 7 and 8 are the narrowest, touched by only 3 colors. Column 9 is the tightest column, with 2 colors.
A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 8 rows. Row 9 is the crowded extreme at 4 colors — worth revisiting later, not now.
Working through the middle game
The middle of #133 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.
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 #133 is rated hard
The hard label comes out of four measurable things about #133: six of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 19. 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.
Stuck on #133 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.
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 #133, start hunting for it near column 9.
A non-spoiling distribution check: solved, #133 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 #133?
Puzzle #133 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and six of its regions are confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #133 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #133, 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 #133?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.