9x9 Queens Puzzle #112
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 21-step reasoning is written out below.
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Step-by-step solution for puzzle #112
What follows is a complete solve of puzzle #112, worked the way a person works it rather than the way a computer does. It runs to 21 deductions — 9 queen placements and 12 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: last square in a region, region shadow, region confined to one line, line confined to one region, pairing argument on columns, proof by contradiction.
Show the full walkthrough (15 steps) — contains the solution
Step 1. Only row 1, column 9 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 9, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. 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. Cross out the whole of row 6 and column 1, the remaining red squares, and the ring of neighbours around row 6, column 1.
Step 3. The amber region has 3 squares left, spread across rows 2 and 3, columns 3 and 5. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.
Step 4. Wherever blue ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Pale yellow cannot leave row 7 — every one of its open squares (2 open squares) sits there. That row belongs to pale yellow, so every square on it owned by another colour is dead: 6 squares in all.
The board after step 5 — 2 of 9 queens placed. Step 6. The khaki region has 3 squares left, spread across rows 4 and 8, columns 3 and 4. 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. Row 5 has only slate squares left (6 open squares). The row needs a queen and only slate can provide one, which pins that colour to this row and kills its squares elsewhere: 6 squares.
Step 8. Every open square left in row 9 (3 open squares) belongs to the orange region. Since that row must contain a queen, orange is the colour that supplies it — so orange squares anywhere else on the board are out, 2 squares in all.
Step 9. Row 8 has only khaki squares left (2 open squares). The row needs a queen and only khaki can provide one, which pins that colour to this row and kills its squares elsewhere: 1 square.
Step 10. 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: 1 square come off.
The board after step 10 — 2 of 9 queens placed. Step 11. Amber, blue, khaki, and orange are now boxed into columns 2–5 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 3 squares at once.
Step 12. All of the slate region's remaining squares (3 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 1 square come off.
Step 13. Test row 2, column 3 for amber: place a queen there, run the forced consequences through, and the khaki region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 14. All of the amber region's remaining squares (2 open squares) lie in column 5. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 15. 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: 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 4; blue has one square left, so its queen takes row 3, column 3; amber has one square left, so its queen takes row 2, column 5; mint has one square left, so its queen takes row 4, column 8; 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 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 7, column 7.
The board after step 15 — 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 #112
This is puzzle number 112 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.
Puzzle #112 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 #112 are: indigo (1 square), red (1 square), orange (6 squares), pale yellow (7 squares), amber (8 squares), blue (10 squares), mint (14 squares), khaki (16 squares), slate (18 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 18 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 1 square while slate has 18 to spare, a gap of 17. 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 1×1 frame; slate is the loosest, occupying only 50% of its 6×6 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 9–9). 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: 6 squares across rows 8–9, columns 2–5.
Pale yellow covers 7 squares, spanning 4 rows and 3 columns (rows 6–9, columns 7–9).
Amber occupies 8 squares within rows 1–3, columns 3–7.
Blue: 10 squares across rows 1–4, columns 1–4.
Mint covers 14 squares, spanning 6 rows and 4 columns (rows 1–6, columns 6–9).
Khaki occupies 16 squares within rows 4–9, columns 1–4.
Slate: 18 squares across rows 4–9, columns 3–8.
Where to start on this board
Begin where the choices are fewest. On this board that means indigo, which owns just 1 square inside a 1×1 window running from row 1 to row 1 and column 9 to column 9. 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. Red 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 #112
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 3, 5 and 7 are the narrowest, touched by only 3 colors. Columns 1, 2, 5, 6, 8 and 9 are the tightest columns, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 8 rows. Row 6 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
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.
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.
Resist the plausible-looking queen. On #112 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 #112 is rated hard
The hard label comes out of four measurable things about #112: four 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 17. Boards that score tight on those measures give up their first queen fast.
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.
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 2 on this board.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 6 queens in the middle, including 1 in a corner. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #112?
Puzzle #112 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #112 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #112, 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 #112?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 9–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.