12x12 Queens Puzzle #14
Difficulty: hard. 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 25-step reasoning is written out below.
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Step-by-step solution for puzzle #14
What follows is a complete solve of puzzle #14, worked the way a person works it rather than the way a computer does. It runs to 25 deductions — 12 queen placements and 13 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, pairing argument on columns, proof by contradiction, last square in a row.
Show the full walkthrough (17 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 rose region is down to one open square, at row 12, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 12 and column 1, the remaining rose squares, and the ring of neighbours around row 12, column 1.
Step 3. The red region has 6 squares left, spread across rows 5–7, columns 2–5. 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 4. All of the orange region's remaining squares (2 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 11 squares come off.
The board after step 4 — 2 of 12 queens placed. Step 5. Red is down to 4 candidates in rows 6 and 7, columns 2, 3, and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 6. All of the teal region's remaining squares (2 open squares) lie in column 3. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 7 squares come off.
Step 7. Slate cannot leave column 2 — every one of its open squares (3 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 4 squares in all.
Step 8. The red region is down to one open square, at row 6, column 5. Every colour holds exactly one queen, so that square is forced. Row 6 and column 5 are now spoken for, along with the squares touching that queen.
Step 9. The blue region has 2 squares left, spread across rows 2 and 3, columns 6 and 8. 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.
The board after step 9 — 3 of 12 queens placed. Step 10. Wherever mint ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. Blue, mint, and khaki are now boxed into columns 6–8 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 6 squares at once.
Step 12. The pale yellow region has 3 squares left, spread across rows 5 and 7, columns 11 and 12. 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 12 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.
The board after step 13 — 3 of 12 queens placed. Step 14. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 10. 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 15. Test row 3, column 11 for amber: place a queen there, run the forced consequences through, and row 4 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 16. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 12. 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. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: row 3 is down to one square, putting a queen on row 3, column 8; row 2 is down to one square, putting a queen on row 2, column 11; row 4 is down to one square, putting a queen on row 4, column 6; khaki has one square left, so its queen takes row 8, column 7; orange has one square left, so its queen takes row 9, column 4; slate has one square left, so its queen takes row 5, column 2; pale yellow has one square left, so its queen takes row 7, column 12; teal has one square left, so its queen takes row 11, column 3; sea green has one square left, so its queen takes row 10, column 10. 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 10, column 10.
The board after step 17 — 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 #14
Board #14 of the 12x12 bank carries a hard rating. Its 144 squares are shared between twelve colored regions, and the solution places twelve queens: one per row, one per column, one per region, none of them side by side or corner to corner.
As with every puzzle in this collection, #14 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 #14 are: indigo (1 square), rose (1 square), orange (3 squares), red (6 squares), mint (11 squares), amber (13 squares), pale yellow (14 squares), teal (15 squares), slate (16 squares), khaki (17 squares), blue (20 squares), sea green (27 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the sea green region is the roomiest area on the grid with 27 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 indigo, with 1 square, is under far more pressure than sea green with 27. A 26-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 indigo is the most solid, filling 100% of its 1×1 box, and mint the most diffuse at 46% of its 6×4 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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 9–9. It never leaves row 1, so its queen is locked to that row.
Rose: 1 square across rows 12–12, columns 1–1. It never leaves row 12, so its queen is locked to that row.
Orange covers 3 squares, spanning 3 rows and 1 column (rows 7–9, columns 4–4). It sits wholly inside column 4, so its queen must be in that column.
Red occupies 6 squares within rows 5–7, columns 2–5.
Mint: 11 squares across rows 2–7, columns 4–7.
Amber covers 13 squares, spanning 5 rows and 4 columns (rows 1–5, columns 9–12).
Pale yellow occupies 14 squares within rows 4–8, columns 7–12.
Teal: 15 squares across rows 8–12, columns 1–6.
Slate covers 16 squares, spanning 9 rows and 3 columns (rows 2–10, columns 1–3).
Khaki occupies 17 squares within rows 6–11, columns 5–8.
Blue: 20 squares across rows 1–4, columns 1–10.
Sea green covers 27 squares, spanning 6 rows and 7 columns (rows 7–12, columns 6–12).
Where to start on this board
Indigo is the natural opening on #14: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 9–9. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.
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. Rose is no more than two rows tall, so its queen is 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 #14
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 11 and 12 are the narrowest, touched by only 3 colors. Columns 11 and 12 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 7, meeting 7 colors, is the opposite case — the busiest line on #14 and the last one you should expect to crack.
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.
The whole middle game reduces to two sweeps. Sweep the twelve colors looking for one with a single square left; sweep the twelve rows and twelve columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 12x12 solve from a stalled one.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #14 is rated hard
Ratings track how generous the opening is. On #14 you get five regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 26-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
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.
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 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.
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 #14, start hunting for it near row 1.
Here is a shape check that gives nothing away: when #14 is solved, 3 queens sit on the outer edge of the grid and 9 sit inside it, with 1 queen in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 #14?
Puzzle #14 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #14 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #14, 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 #14?
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.