9x9 Queens Puzzle #125
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 24-step reasoning is written out below.
Previous puzzle · All 9x9 puzzles · Next puzzle
Step-by-step solution for puzzle #125
What follows is a complete solve of puzzle #125, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 9 queen placements and 15 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 shadow, region confined to one line, proof by contradiction, pairing argument on rows, last square in a region.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Wherever amber ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 2. Khaki is down to 4 candidates in rows 7 and 8, columns 1–3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 3. Orange cannot leave row 9 — every one of its open squares (4 open squares) sits there. That row belongs to orange, so every square on it owned by another colour is dead: 5 squares in all.
Step 4. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 1. Follow the forced moves that would create and column 7 runs out of open squares entirely — impossible. So that square is ruled out for good.
The board after step 4 — 0 of 9 queens placed. Step 5. Test row 3, column 1 for indigo: place a queen there, run the forced consequences through, and the red region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 6. The indigo region has 3 squares left, spread across rows 1–3, columns 1 and 2. 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 7. Test row 3, column 2 for indigo: place a queen there, run the forced consequences through, and the red region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 8. Indigo is down to 2 candidates in rows 1 and 2, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 9. Indigo and amber are now boxed into rows 1 and 2 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 7 squares at once.
The board after step 9 — 0 of 9 queens placed. Step 10. Wherever blue ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 11. Test row 1, column 5 for amber: place a queen there, run the forced consequences through, and the red region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 12. The amber region has 3 squares left, spread across rows 1 and 2, columns 6 and 7. 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 13. Test row 2, column 2 for indigo: place a queen there, run the forced consequences through, and the orange region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 13 — 0 of 9 queens placed. Step 14. The indigo region is down to one open square, at row 1, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 1, the remaining indigo squares, and the ring of neighbours around row 1, column 1.
Step 15. Amber cannot leave row 2 — every one of its open squares (2 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 1 square in all.
Step 16. All of the khaki region's remaining squares (2 open squares) lie in row 8. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 9 squares come off.
Step 17. 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: orange has one square left, so its queen takes row 9, column 4; blue has one square left, so its queen takes row 3, column 5; amber has one square left, so its queen takes row 2, column 7; mint has one square left, so its queen takes row 4, column 9; khaki has one square left, so its queen takes row 8, column 2; slate has one square left, so its queen takes row 5, column 3; red has one square left, so its queen takes row 6, column 6; pale yellow has one square left, so its queen takes row 7, 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 7, column 8.
The board after step 17 — 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 #125
Puzzle #125 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.
Puzzle #125 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #125 are: amber (4 squares), khaki (4 squares), orange (4 squares), indigo (5 squares), blue (10 squares), red (11 squares), slate (13 squares), mint (14 squares), pale yellow (16 squares). An even split would hand every color 9 squares, so the amber region is running 5 squares under that average while the pale yellow region is the roomiest area on the grid with 16 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 amber, with 4 squares, is under far more pressure than pale yellow with 16. A 12-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 orange is the most solid, filling 100% of its 1×4 box, and mint the most diffuse at 47% of its 6×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Here is the full footprint of every color on puzzle #125, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Amber occupies 4 squares within rows 1–2, columns 5–7.
Khaki: 4 squares across rows 7–8, columns 1–3.
Orange covers 4 squares, spanning 1 row and 4 columns (rows 9–9, columns 1–4). It never leaves row 9, so its queen is locked to that row.
Indigo occupies 5 squares within rows 1–3, columns 1–2.
Blue: 10 squares across rows 1–4, columns 2–6.
Red covers 11 squares, spanning 5 rows and 4 columns (rows 5–9, columns 3–6).
Slate occupies 13 squares within rows 3–7, columns 1–5.
Mint: 14 squares across rows 1–6, columns 5–9.
Pale yellow covers 16 squares, spanning 6 rows and 4 columns (rows 4–9, columns 6–9).
Where to start on this board
The most constrained color is where a solve should begin, and on #125 that is amber with 4 squares. It reaches from row 1 to row 2 and from column 5 to column 7 — 2 rows by 3 columns. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
With 2 rows and 3 columns of room, amber needs narrowing rather than solving outright. Test its 4 squares against the neighbours: a candidate that would strand another color is dead, and eliminating even two of them tightens the whole corner of the board around rows 1–2.
Work outward to the other flat regions next. Orange and khaki 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 #125
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 8 and 9 are the narrowest, touched by only 3 colors. Columns 8 and 9 are the tightest columns, 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 7 is the crowded extreme at 4 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.
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.
A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.
Why puzzle #125 is rated hard
The rating reflects how much the board concedes up front. Puzzle #125 has one region confined to a single row or column, its smallest color holds 4 squares, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 12 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
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 #125, start hunting for it near column 8.
A non-spoiling distribution check: solved, #125 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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 9x9 Queens puzzle #125?
Puzzle #125 is rated hard. Its smallest color region holds 4 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #125 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #125, 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 #125?
Start with the amber region — the most constrained color on this board, with 4 squares spanning rows 1–2 and columns 5–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.