12x12 Queens Puzzle #71
Difficulty: medium. Place 12 queens on this 12x12 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 27-step reasoning is written out below.
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Step-by-step solution for puzzle #71
What follows is a complete solve of puzzle #71, worked the way a person works it rather than the way a computer does. It runs to 27 deductions — 12 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: last square in a region, region confined to one line, pairing argument on columns, line confined to one region, last square in a row, region shadow, proof by contradiction.
Show the full walkthrough (20 steps) — contains the solution
Step 1. Only row 1, column 1 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 1, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The pale yellow region is down to one open square, at row 7, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 3, the remaining pale yellow squares, and the ring of neighbours around row 7, column 3.
Step 3. Red cannot leave column 12 — 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: 10 squares in all.
Step 4. All of the sea green region's remaining squares (2 open squares) lie in column 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 12 squares come off.
Step 5. Blue, slate, and teal are now boxed into columns 2, 4, and 5 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 13 squares at once.
The board after step 5 — 2 of 12 queens placed. Step 6. Every open square left in row 2 (5 open squares) belongs to the amber region. Since that row must contain a queen, amber is the colour that supplies it — so amber squares anywhere else on the board are out, 7 squares in all.
Step 7. Look along row 3: one square survives, row 3, column 2, which belongs to blue. A row cannot be left empty, so the queen goes there. That clears row 3, column 2, the rest of the blue region and every square adjacent to it — diagonals included.
Step 8. Slate is down to 2 candidates in rows 5 and 8, columns 4 and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.
Step 9. Teal cannot leave row 11 — every one of its open squares (2 open squares) sits there. That row belongs to teal, so every square on it owned by another colour is dead: 4 squares in all.
Step 10. The sea green region is down to one open square, at row 10, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 10, the remaining sea green squares, and the ring of neighbours around row 10, column 10.
The board after step 10 — 4 of 12 queens placed. Step 11. Test row 2, column 6 for amber: 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.
Step 12. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 7. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 13. Test row 2, column 8 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 14. Now a short proof by contradiction. Suppose the mint queen went to row 5, column 6. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 15. Test row 5, column 7 for mint: 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.
The board after step 15 — 4 of 12 queens placed. Step 16. Now a short proof by contradiction. Suppose the mint queen went to row 5, column 8. Follow the forced moves that would create and row 4 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 17. Mint and rose are now boxed into columns 6 and 7 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 1 square at once.
Step 18. Now a short proof by contradiction. Suppose the mint queen went to row 6, column 6. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 19. Test row 6, column 7 for mint: 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 20. 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: mint has one square left, so its queen takes row 4, column 6; row 9 is down to one square, putting a queen on row 9, column 8; rose has one square left, so its queen takes row 12, column 7; khaki has one square left, so its queen takes row 8, column 11; amber has one square left, so its queen takes row 2, column 9; slate has one square left, so its queen takes row 5, column 4; red has one square left, so its queen takes row 6, column 12; teal has one square left, so its queen takes row 11, column 5. 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 11, column 5.
The board after step 20 — 12 of 12 queens placed.
That completes the board: 12 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 12x12 Queens puzzle #71
12x12 puzzle #71 is rated medium. Twelve colored regions cover its 144 squares, and your job is to seat twelve queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Like every board on this site, #71 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.
How the colors divide this board
From smallest to largest, the regions on puzzle #71 are: indigo (1 square), pale yellow (1 square), sea green (2 squares), red (3 squares), teal (10 squares), blue (12 squares), orange (15 squares), rose (16 squares), mint (18 squares), khaki (20 squares), slate (21 squares), amber (25 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the amber region is the roomiest area on the grid with 25 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 amber has 25 to spare, a gap of 24. Attack the cramped end and the roomy regions tend to resolve themselves.
Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×1 box that encloses it, while teal is the most scattered at 50% of its 4×5 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.
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 occupies 1 square within rows 1–1, columns 1–1. It never leaves row 1, so its queen is locked to that row.
Pale yellow: 1 square across rows 7–7, columns 3–3. It never leaves row 7, so its queen is locked to that row.
Sea green covers 2 squares, spanning 2 rows and 1 column (rows 10–11, columns 10–10). It sits wholly inside column 10, so its queen must be in that column.
Red occupies 3 squares within rows 5–7, columns 12–12. It sits wholly inside column 12, so its queen must be in that column.
Teal: 10 squares across rows 9–12, columns 1–5.
Blue covers 12 squares, spanning 5 rows and 4 columns (rows 1–5, columns 1–4).
Orange occupies 15 squares within rows 4–11, columns 7–9.
Rose: 16 squares across rows 8–12, columns 2–7.
Mint covers 18 squares, spanning 7 rows and 5 columns (rows 1–7, columns 4–8).
Khaki occupies 20 squares within rows 5–12, columns 8–12.
Slate: 21 squares across rows 3–10, columns 1–5.
Amber covers 25 squares, spanning 6 rows and 7 columns (rows 1–6, columns 6–12).
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 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 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. Pale yellow and sea green 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 #71
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2 and 12 are the narrowest, touched by only 3 colors. Column 11 is the tightest column, with 2 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 11 rows. Row 5 sits at the far end with 7 colors crossing it; it will almost certainly be among the last to settle.
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.
Keep alternating between two views of the board. In the region view you ask which of the twelve colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 12x12 board unwinds.
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 #71 is rated medium
Ratings track how generous the opening is. On #71 you get six regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 24-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
If #71 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.
Techniques that work well at 12x12
At 12x12 you are managing 144 squares, and holding the whole board in your head will not work. Split it into quadrants and solve locally. Most useful deductions on a grid this large are regional — a color pinched between two placed queens, or a band of rows that two regions must share. Trust the marks on the board over your memory.
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 column 11 on this board.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 9 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 12x12 Queens puzzle #71?
Puzzle #71 is rated medium. 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 12x12 puzzle #71 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #71, has been verified to have exactly one valid arrangement of twelve queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #71?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.