12x12 Queens Puzzle #15
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 17-step reasoning is written out below.
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Step-by-step solution for puzzle #15
What follows is a complete solve of puzzle #15, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 12 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 shadow.
Techniques this board needs: last square in a region, region confined to one line, region shadow.
Show the full walkthrough (10 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 7; red has one square left, so its queen takes row 6, column 3; khaki has one square left, so its queen takes row 8, column 12; amber has one square left, so its queen takes row 2, column 11. 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 11.
Step 2. Slate cannot leave column 10 — 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: 7 squares in all.
Step 3. The orange region has 4 squares left, spread across rows 7, 9, and 12, columns 8 and 9. 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.
The board after step 3 — 4 of 12 queens placed. Step 4. Sea green cannot leave column 1 — every one of its open squares (4 open squares) sits there. That column belongs to sea green, so every square in it owned by another colour is dead: 4 squares in all.
Step 5. The rose region is down to one open square, at row 12, column 2. Every colour holds exactly one queen, so that square is forced. Row 12 and column 2 are now spoken for, along with the squares touching that queen.
Step 6. Only row 4, column 4 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 4, the rest of the mint region and every square adjacent to it — diagonals included.
Step 7. The slate region is down to one open square, at row 5, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 10, the remaining slate squares, and the ring of neighbours around row 5, column 10.
The board after step 7 — 7 of 12 queens placed. Step 8. Blue cannot leave row 3 — every one of its open squares (2 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 1 square in all.
Step 9. All of the orange region's remaining squares (2 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 10. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: blue has one square left, so its queen takes row 3, column 8; pale yellow has one square left, so its queen takes row 7, column 5; orange has one square left, so its queen takes row 9, column 9; sea green has one square left, so its queen takes row 10, column 1; teal has one square left, so its queen takes row 11, column 6. 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 6.
The board after step 10 — 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 #15
12x12 puzzle #15 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, #15 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 #15 are: indigo (1 square), red (1 square), khaki (1 square), rose (3 squares), amber (6 squares), sea green (6 squares), slate (10 squares), blue (15 squares), mint (20 squares), teal (23 squares), orange (24 squares), pale yellow (34 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the pale yellow region is the roomiest area on the grid with 34 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 pale yellow has 34 to spare, a gap of 33. Attack the cramped end and the roomy regions tend to resolve themselves.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and teal the most diffuse at 48% of its 6×8 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #15 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Indigo: 1 square across rows 1–1, columns 7–7. It never leaves row 1, so its queen is locked to that row.
Red covers 1 square, spanning 1 row and 1 column (rows 6–6, columns 3–3). It never leaves row 6, so its queen is locked to that row.
Khaki occupies 1 square within rows 8–8, columns 12–12. It never leaves row 8, so its queen is locked to that row.
Rose: 3 squares across rows 12–12, columns 1–3. It never leaves row 12, so its queen is locked to that row.
Amber covers 6 squares, spanning 3 rows and 3 columns (rows 1–3, columns 10–12).
Sea green occupies 6 squares within rows 6–11, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Slate: 10 squares across rows 3–6, columns 10–12.
Blue covers 15 squares, spanning 5 rows and 4 columns (rows 1–5, columns 7–10).
Mint occupies 20 squares within rows 1–6, columns 1–4.
Teal: 23 squares across rows 7–12, columns 2–9.
Orange covers 24 squares, spanning 7 rows and 5 columns (rows 6–12, columns 8–12).
Pale yellow occupies 34 squares within rows 1–10, columns 2–8.
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 7 to column 7. 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, khaki and rose 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 #15
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 11 and 12 are the narrowest, touched by only 3 colors. Columns 5 and 6 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 11 rows. Row 6 is the crowded extreme at 6 colors — worth revisiting later, not now.
Working through the middle game
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
With twelve queens to seat and twelve regions competing over 144 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 12x12 boards quickly and players who stall halfway.
Resist the plausible-looking queen. On #15 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 #15 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers eight single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 33. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
If #15 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 5 on this board.
A non-spoiling distribution check: solved, #15 carries 4 queens on the perimeter against 8 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 12x12 Queens puzzle #15?
Puzzle #15 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and eight of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #15 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #15, 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 #15?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.