9x9 Queens Puzzle #120
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 14-step reasoning is written out below.
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Step-by-step solution for puzzle #120
What follows is a complete solve of puzzle #120, worked the way a person works it rather than the way a computer does. It runs to 14 deductions — 9 queen placements and 5 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.
Show the full walkthrough (8 steps) — contains the solution
Step 1. Only row 8, column 1 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 1, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 2. The blue region is down to one open square, at row 3, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 2, the remaining blue squares, and the ring of neighbours around row 3, column 2.
Step 3. Indigo cannot leave row 1 — every one of its open squares (3 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 5 squares in all.
Step 4. All of the mint region's remaining squares (2 open squares) lie in row 4. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.
The board after step 4 — 2 of 9 queens placed. Step 5. Red cannot leave column 3 — every one of its open squares (2 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 4 squares in all.
Step 6. All of the amber region's remaining squares (3 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 7. Slate cannot leave row 5 — every one of its open squares (2 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 4 squares in all.
Step 8. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: red has one square left, so its queen takes row 6, column 3; pale yellow has one square left, so its queen takes row 7, column 7; slate has one square left, so its queen takes row 5, column 8; indigo has one square left, so its queen takes row 1, column 6; amber has one square left, so its queen takes row 2, column 4; mint has one square left, so its queen takes row 4, column 5; orange has one square left, so its queen takes row 9, 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 9, column 9.
The board after step 8 — 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 #120
This is puzzle number 120 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, #120 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 #120 are: khaki (1 square), indigo (3 squares), blue (3 squares), mint (5 squares), pale yellow (8 squares), slate (11 squares), red (14 squares), amber (17 squares), orange (19 squares). An even split would hand every color 9 squares, so the khaki region is running 8 squares under that average while the orange region is the roomiest area on the grid with 19 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Khaki is working with 1 square while orange has 19 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 1×3 frame; pale yellow 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.
Khaki: 1 square across rows 8–8, columns 1–1. It never leaves row 8, so its queen is locked to that row.
Indigo covers 3 squares, spanning 1 row and 3 columns (rows 1–1, columns 6–8). It never leaves row 1, so its queen is locked to that row.
Blue occupies 3 squares within rows 3–4, columns 1–2.
Mint: 5 squares across rows 2–4, columns 5–7.
Pale yellow covers 8 squares, spanning 3 rows and 6 columns (rows 5–7, columns 4–9).
Slate occupies 11 squares within rows 1–5, columns 6–9.
Red: 14 squares across rows 4–9, columns 1–3.
Amber covers 17 squares, spanning 5 rows and 6 columns (rows 1–5, columns 1–6).
Orange occupies 19 squares within rows 7–9, columns 3–9.
Where to start on this board
Begin where the choices are fewest. On this board that means khaki, which owns just 1 square inside a 1×1 window running from row 8 to row 8 and column 1 to column 1. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because khaki never leaves row 8, its queen must be in that row — and that instantly settles row 8 for everyone else. Sweep along it and X out every square that is not khaki. 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. Indigo and blue 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 #120
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 6 and 9 are the narrowest, touched by only 2 colors. Columns 2, 3, 4 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 4 is the crowded extreme at 5 colors — worth revisiting later, not now.
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 #120 is rated hard
Ratings track how generous the opening is. On #120 you get three 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 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.
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 6 on this board.
A non-spoiling distribution check: solved, #120 carries 3 queens on the perimeter against 6 queens inside, 1 of them cornered. 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 #120?
Puzzle #120 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #120 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #120, 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 #120?
Start with the khaki region — the most constrained color on this board, with 1 square spanning rows 8–8 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.