12x12 Queens Puzzle #85
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 20-step reasoning is written out below.
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Step-by-step solution for puzzle #85
What follows is a complete solve of puzzle #85, worked the way a person works it rather than the way a computer does. It runs to 20 deductions — 12 queen placements and 8 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: region confined to one line, last square in a region, region shadow, last square in a column.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Teal cannot leave column 4 — every one of its open squares (2 open squares) sits there. That column belongs to teal, so every square in it owned by another colour is dead: 14 squares in all.
Step 2. 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. Cross out the whole of row 12 and column 2, the remaining rose squares, and the ring of neighbours around row 12, column 2.
Step 3. Only row 11, column 4 is still open in teal, and since each colour must take a queen, that settles it. Everything else in row 11, in column 4, in teal and in the ring around that square is now dead.
Step 4. The sea green region is down to one open square, at row 10, column 1. Every colour holds exactly one queen, so that square is forced. Row 10 and column 1 are now spoken for, along with the squares touching that queen.
The board after step 4 — 3 of 12 queens placed. Step 5. Khaki cannot leave column 3 — every one of its open squares (2 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 7 squares in all.
Step 6. 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. Cross out the whole of row 6 and column 5, the remaining red squares, and the ring of neighbours around row 6, column 5.
Step 7. Orange cannot leave row 9 — every one of its open squares (2 open squares) sits there. That row belongs to orange, so every square on it owned by another colour is dead: 8 squares in all.
Step 8. The khaki region is down to one open square, at row 8, column 3. Every colour holds exactly one queen, so that square is forced. Row 8 and column 3 are now spoken for, along with the squares touching that queen.
Step 9. Pale yellow cannot leave row 7 — every one of its open squares (3 open squares) sits there. That row belongs to pale yellow, so every square on it owned by another colour is dead: 3 squares in all.
The board after step 9 — 5 of 12 queens placed. Step 10. The slate region is down to one open square, at row 5, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 7, the remaining slate squares, and the ring of neighbours around row 5, column 7.
Step 11. Only row 9, column 6 is still open in orange, and since each colour must take a queen, that settles it. Everything else in row 9, in column 6, in orange and in the ring around that square is now dead.
Step 12. All of the indigo region's remaining squares (2 open squares) lie in column 8. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 13. Wherever blue ends up among its 6 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 13 — 7 of 12 queens placed. Step 14. All of the mint 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: 5 squares come off.
Step 15. Only row 1, column 8 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 8, in indigo and in the ring around that square is now dead.
Step 16. All of the amber region's remaining squares (2 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 17. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: column 11 is down to one square, putting a queen on row 7, column 11; column 10 is down to one square, putting a queen on row 2, column 10; mint has one square left, so its queen takes row 4, column 9; blue has one square left, so its queen takes row 3, column 12. 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 3, column 12.
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 #85
Puzzle #85 is one of the hard-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, #85 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 #85 are: teal (2 squares), rose (3 squares), red (5 squares), amber (6 squares), sea green (6 squares), blue (9 squares), mint (9 squares), pale yellow (15 squares), khaki (15 squares), indigo (20 squares), orange (25 squares), slate (29 squares). An even split would hand every color 12 squares, so the teal region is running 10 squares under that average while the slate 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 27 square difference between teal and slate 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. Teal is the tidiest shape on #85, packing 100% of its 2×1 bounding box, whereas slate straggles across its 10×9 box at just 32%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Below is every color on #85 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.
Teal covers 2 squares, spanning 2 rows and 1 column (rows 11–12, columns 4–4). It sits wholly inside column 4, so its queen must be in that column.
Rose occupies 3 squares within rows 11–12, columns 2–3.
Red: 5 squares across rows 6–7, columns 3–5.
Amber covers 6 squares, spanning 2 rows and 4 columns (rows 1–2, columns 9–12).
Sea green occupies 6 squares within rows 10–12, columns 1–3.
Blue: 9 squares across rows 2–7, columns 10–12.
Mint covers 9 squares, spanning 4 rows and 4 columns (rows 2–5, columns 8–11).
Pale yellow occupies 15 squares within rows 6–10, columns 8–12.
Khaki: 15 squares across rows 6–10, columns 1–6.
Indigo covers 20 squares, spanning 4 rows and 7 columns (rows 1–4, columns 2–8).
Orange occupies 25 squares within rows 7–12, columns 5–12.
Slate: 29 squares across rows 1–10, columns 1–9.
Where to start on this board
Open on the tightest color. Here that is teal, confined to 2 squares between rows 11 and 12 and columns 4 and 4. Its queen has nowhere else to go, which means those 2 rows and 1 column carry more constraint than anywhere else on #85.
Because teal stays inside column 4, its queen belongs to that column and no other region may use it. Cross out the rest of column 4 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
Work outward to the other flat regions next. Amber, red 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 #85
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 5 are the narrowest, touched by only 3 colors. Columns 1 and 7 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 7 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
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.
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.
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 #85 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers one single-line region, a 2-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 27. 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.
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 #85 is row 1.
As a sanity check rather than a spoiler: the finished solution to #85 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.
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Questions about this puzzle
How hard is 12x12 Queens puzzle #85?
Puzzle #85 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #85 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #85, 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 #85?
Start with the teal region — the most constrained color on this board, with 2 squares spanning rows 11–12 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.