12x12 Queens Puzzle #87
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 20-step reasoning is written out below.
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Step-by-step solution for puzzle #87
What follows is a complete solve of puzzle #87, 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 pairing argument on rows.
Techniques this board needs: last square in a region, region shadow, region confined to one line, pairing argument on rows.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Only row 5, column 3 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 3, the rest of the slate region and every square adjacent to it — diagonals included.
Step 2. Amber is down to 4 candidates in rows 1 and 2, columns 8–10, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about amber itself.
Step 3. Mint cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 9 squares in all.
Step 4. Wherever red ends up among its 2 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Teal is down to 3 candidates in rows 11 and 12, columns 8 and 9, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about teal itself.
The board after step 5 — 1 of 12 queens placed. Step 6. Here is the pairing argument. The teal and rose regions have no open squares outside rows 11 and 12 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 10 squares come off in one stroke.
Step 7. Khaki cannot leave column 2 — every one of its open squares (3 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 4 squares in all.
Step 8. 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: red has one square left, so its queen takes row 6, column 5; indigo has one square left, so its queen takes row 1, column 4; amber has one square left, so its queen takes row 2, column 10; mint has one square left, so its queen takes row 4, column 1; 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.
Step 9. Rose cannot leave column 11 — every one of its open squares (2 open squares) sits there. That column belongs to rose, so every square in it owned by another colour is dead: 4 squares in all.
Step 10. The orange region is down to one open square, at row 9, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 9, the remaining orange squares, and the ring of neighbours around row 9, column 9.
Step 11. Only row 11, column 8 is still open in teal, and since each colour must take a queen, that settles it. Everything else in row 11, in column 8, in teal and in the ring around that square is now dead.
The board after step 11 — 8 of 12 queens placed. Step 12. The rose region is down to one open square, at row 12, column 11. Every colour holds exactly one queen, so that square is forced. Row 12 and column 11 are now spoken for, along with the squares touching that queen.
Step 13. Pale yellow is down to 2 candidates in rows 7 and 8, columns 6 and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 14. The sea green region is down to one open square, at row 10, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 6, the remaining sea green squares, and the ring of neighbours around row 10, column 6.
Step 15. Only row 7, column 7 is still open in pale yellow, and since each colour must take a queen, that settles it. Everything else in row 7, in column 7, in pale yellow and in the ring around that square is now dead.
Step 16. The khaki region is down to one open square, at row 8, column 2. Every colour holds exactly one queen, so that square is forced. Row 8 and column 2 are now spoken for, along with the squares touching that queen.
The board after step 16 — 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 #87
12x12 puzzle #87 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, #87 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 #87 are: slate (1 square), teal (3 squares), amber (4 squares), rose (5 squares), mint (8 squares), red (8 squares), indigo (11 squares), blue (13 squares), khaki (17 squares), sea green (17 squares), orange (18 squares), pale yellow (39 squares). An even split would hand every color 12 squares, so the slate region is running 11 squares under that average while the pale yellow region is the roomiest area on the grid with 39 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Slate is working with 1 square while pale yellow has 39 to spare, a gap of 38. 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 slate is the most solid, filling 100% of its 1×1 box, and red the most diffuse at 40% of its 5×4 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a medium board.
Slate: 1 square across rows 5–5, columns 3–3. It never leaves row 5, so its queen is locked to that row.
Teal covers 3 squares, spanning 2 rows and 2 columns (rows 11–12, columns 8–9).
Amber occupies 4 squares within rows 1–2, columns 8–10.
Rose: 5 squares across rows 11–12, columns 10–12.
Mint covers 8 squares, spanning 4 rows and 4 columns (rows 3–6, columns 1–4).
Red occupies 8 squares within rows 5–9, columns 2–5.
Indigo: 11 squares across rows 1–3, columns 1–5.
Blue covers 13 squares, spanning 5 rows and 3 columns (rows 1–5, columns 10–12).
Khaki occupies 17 squares within rows 5–12, columns 1–4.
Sea green: 17 squares across rows 8–12, columns 4–8.
Orange covers 18 squares, spanning 6 rows and 4 columns (rows 6–11, columns 9–12).
Pale yellow occupies 39 squares within rows 1–9, columns 4–9.
Where to start on this board
Begin where the choices are fewest. On this board that means slate, which owns just 1 square inside a 1×1 window running from row 5 to row 5 and column 3 to column 3. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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, teal 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 #87
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 4 and 10 are the narrowest, touched by only 3 colors. Columns 6 and 7 are the tightest columns, with 2 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 5 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
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.
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 #87 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 #87 is rated medium
The medium label comes out of four measurable things about #87: two of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 38. Boards that score tight on those measures give up their first queen fast.
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.
If #87 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 6 on this board.
As a sanity check rather than a spoiler: the finished solution to #87 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 #87?
Puzzle #87 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #87 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #87, 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 #87?
Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.