12x12 Queens Puzzle #81
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 18-step reasoning is written out below.
Previous puzzle · All 12x12 puzzles · Next puzzle
Step-by-step solution for puzzle #81
What follows is a complete solve of puzzle #81, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 12 queen placements and 6 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 (13 steps) — contains the solution
Step 1. Only row 12, column 5 is still open in rose, and since each colour must take a queen, that settles it. That clears row 12, column 5, the rest of the rose region and every square adjacent to it — diagonals included.
Step 2. The teal region is down to one open square, at row 11, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 1, the remaining teal squares, and the ring of neighbours around row 11, column 1.
Step 3. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 8 squares in all.
Step 4. The amber region is down to one open square, at row 2, column 10. Every colour holds exactly one queen, so that square is forced. Row 2 and column 10 are now spoken for, along with the squares touching that queen.
The board after step 4 — 3 of 12 queens placed. Step 5. 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: 7 squares in all.
Step 6. The mint region is down to one open square, at row 4, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 2, the remaining mint squares, and the ring of neighbours around row 4, column 2.
Step 7. Only row 5, column 12 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 12, in slate and in the ring around that square is now dead.
Step 8. Orange is down to 3 candidates in rows 9 and 10, columns 3 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
Step 9. Sea green cannot leave row 10 — every one of its open squares (2 open squares) sits there. That row belongs to sea green, so every square on it owned by another colour is dead: 5 squares in all.
The board after step 9 — 5 of 12 queens placed. Step 10. The orange region is down to one open square, at row 9, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 3, the remaining orange squares, and the ring of neighbours around row 9, column 3.
Step 11. Pale yellow is down to 3 candidates in rows 7 and 8, columns 6 and 7, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 12. All of the red 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: 1 square come off.
Step 13. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: indigo has one square left, so its queen takes row 1, column 6; pale yellow has one square left, so its queen takes row 7, column 7; blue has one square left, so its queen takes row 3, column 8; red has one square left, so its queen takes row 6, column 4; sea green has one square left, so its queen takes row 10, column 11; khaki has one square left, so its queen takes row 8, column 9. 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 8, column 9.
The board after step 13 — 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 #81
Puzzle #81 is one of the medium-rated boards in the 12x12 set. Twelve colors divide its 144 squares, and you win by placing twelve queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
As with every puzzle in this collection, #81 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 #81 are: rose (1 square), amber (2 squares), indigo (6 squares), teal (6 squares), slate (9 squares), blue (10 squares), orange (10 squares), pale yellow (13 squares), mint (14 squares), sea green (16 squares), red (28 squares), khaki (29 squares). An even split would hand every color 12 squares, so the rose region is running 11 squares under that average while the khaki region is the roomiest area on the grid with 29 squares.
Every color takes one queen and one only, whatever its size — which is why the 28 square difference between rose and khaki is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
Square count is only half the story — how a color is arranged counts too. Amber is the tidiest shape on #81, packing 100% of its 2×1 bounding box, whereas red straggles across its 9×7 box at just 44%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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.
Rose: 1 square across rows 12–12, columns 5–5. It never leaves row 12, so its queen is locked to that row.
Amber covers 2 squares, spanning 2 rows and 1 column (rows 1–2, columns 10–10). It sits wholly inside column 10, so its queen must be in that column.
Indigo occupies 6 squares within rows 1–4, columns 4–6.
Teal: 6 squares across rows 11–12, columns 1–4.
Slate covers 9 squares, spanning 5 rows and 3 columns (rows 1–5, columns 10–12).
Blue occupies 10 squares within rows 1–3, columns 6–9.
Orange: 10 squares across rows 8–11, columns 2–5.
Pale yellow covers 13 squares, spanning 6 rows and 4 columns (rows 7–12, columns 5–8).
Mint occupies 14 squares within rows 1–7, columns 1–3.
Sea green: 16 squares across rows 9–12, columns 8–12.
Red covers 28 squares, spanning 9 rows and 7 columns (rows 2–10, columns 1–7).
Khaki occupies 29 squares within rows 4–10, columns 7–12.
Where to start on this board
Open on the tightest color. Here that is rose, confined to 1 square between rows 12 and 12 and columns 5 and 5. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #81.
Because rose never leaves row 12, its queen must be in that row — and that instantly settles row 12 for everyone else. Sweep along it and X out every square that is not rose. 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. Amber and teal 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 #81
A quick way to find the next move is to count how many different colors touch each line. On this board, row 6 is the narrowest, touched by only 3 colors. Columns 1, 9, 11 and 12 are the tightest columns, with 3 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 11 rows. By contrast row 2 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
Working through the middle game
The middle of #81 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.
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 #81 is rated medium
Ratings track how generous the opening is. On #81 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 28-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
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.
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.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #81 is row 6.
As a sanity check rather than a spoiler: the finished solution to #81 puts 4 queens on the outer border and 8 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 12x12 Queens puzzle #81?
Puzzle #81 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #81 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #81, 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 #81?
Start with the rose region — the most constrained color on this board, with 1 square spanning rows 12–12 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.