12x12 Queens Puzzle #84
Difficulty: easy. 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 #84
What follows is a complete solve of puzzle #84, 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, line confined to one region, last square in a row, pairing argument on rows, last square in a column.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Only row 9, column 12 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 12, the rest of the orange region and every square adjacent to it — diagonals included.
Step 2. Amber is down to 3 candidates in rows 1 and 2, columns 4 and 5, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about amber itself.
Step 3. Rose cannot leave row 12 — every one of its open squares (2 open squares) sits there. That row belongs to rose, so every square on it owned by another colour is dead: 11 squares in all.
Step 4. Every open square left in row 3 (11 open squares) belongs to the blue region. Since that row must contain a queen, blue is the colour that supplies it — so blue squares anywhere else on the board are out, 33 squares in all.
Step 5. Look along row 2: one square survives, row 2, column 4, which belongs to amber. A row cannot be left empty, so the queen goes there. Everything else in row 2, in column 4, in amber and in the ring around that square is now dead.
The board after step 5 — 2 of 12 queens placed. Step 6. Every open square left in row 1 (4 open squares) belongs to the indigo region. Since that row must contain a queen, indigo is the colour that supplies it — so indigo squares anywhere else on the board are out, 1 square in all.
Step 7. Column 5 has only sea green squares left (2 open squares). The column needs a queen and only sea green can provide one, which pins that colour to this column and kills its squares elsewhere: 5 squares.
Step 8. Every open square left in column 3 (3 open squares) belongs to the pale yellow region. Since that column must contain a queen, pale yellow is the colour that supplies it — so pale yellow squares anywhere else on the board are out, 3 squares in all.
Step 9. Khaki, sea green, and teal are now boxed into rows 8, 10, and 11 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 7 squares at once.
The board after step 9 — 2 of 12 queens placed. Step 10. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: pale yellow has one square left, so its queen takes row 7, column 3; red has one square left, so its queen takes row 6, column 1; column 2 is down to one square, putting a queen on row 3, column 2; column 6 is down to one square, putting a queen on row 12, column 6; sea green has one square left, so its queen takes row 10, column 5; column 7 is down to one square, putting a queen on row 8, column 7; teal has one square left, so its queen takes row 11, 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 11, column 11.
Step 11. Mint cannot leave column 10 — 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: 3 squares in all.
Step 12. The slate region is down to one open square, at row 5, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 8, the remaining slate squares, and the ring of neighbours around row 5, column 8.
Step 13. Only row 1, column 9 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 9, in indigo and in the ring around that square is now dead.
Step 14. The mint region is down to one open square, at row 4, column 10. Every colour holds exactly one queen, so that square is forced. Row 4 and column 10 are now spoken for, along with the squares touching that queen.
The board after step 14 — 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 #84
This is puzzle number 84 in the 12x12 collection, rated easy. The grid holds 144 squares split between twelve colored regions, and a finished board carries twelve queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
Puzzle #84 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #84 are: orange (1 square), rose (2 squares), amber (3 squares), indigo (9 squares), mint (9 squares), red (9 squares), slate (10 squares), khaki (10 squares), pale yellow (13 squares), sea green (14 squares), teal (18 squares), blue (46 squares). An even split would hand every color 12 squares, so the orange region is running 11 squares under that average while the blue region is the roomiest area on the grid with 46 squares.
Every color takes one queen and one only, whatever its size — which is why the 45 square difference between orange and blue 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.
Shape matters as much as size. The orange region is the most compact color here, filling 100% of the 1×1 box that encloses it, while indigo is the most scattered at 45% of its 4×5 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.
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.
Orange: 1 square across rows 9–9, columns 12–12. It never leaves row 9, so its queen is locked to that row.
Rose covers 2 squares, spanning 1 row and 2 columns (rows 12–12, columns 6–7). It never leaves row 12, so its queen is locked to that row.
Amber occupies 3 squares within rows 1–2, columns 4–5.
Indigo: 9 squares across rows 1–4, columns 8–12.
Mint covers 9 squares, spanning 4 rows and 3 columns (rows 4–7, columns 10–12).
Red occupies 9 squares within rows 6–12, columns 1–2.
Slate: 10 squares across rows 4–8, columns 7–9.
Khaki covers 10 squares, spanning 4 rows and 4 columns (rows 8–11, columns 7–10).
Pale yellow occupies 13 squares within rows 6–10, columns 2–4.
Sea green: 14 squares across rows 6–12, columns 3–6.
Teal covers 18 squares, spanning 6 rows and 5 columns (rows 7–12, columns 8–12).
Blue occupies 46 squares within rows 1–8, columns 1–11.
Where to start on this board
The most constrained color is where a solve should begin, and on #84 that is orange with 1 square. It reaches from row 9 to row 9 and from column 12 to column 12 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
Because orange never leaves row 9, its queen must be in that row — and that instantly settles row 9 for everyone else. Sweep along it and X out every square that is not orange. 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 and amber 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 #84
A quick way to find the next move is to count how many different colors touch each line. On this board, row 3 is the narrowest, touched by only 2 colors. Column 1 is the tightest column, with 2 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 colors, so one of that handful must take the row and the rest are squeezed into the other 11 rows. Row 8 sits at the far end with 7 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
The middle of #84 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.
Two checks, run alternately, will carry you through the 144 squares of this grid. First: has any of the twelve regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 12x12 boards almost always run one of those checks and forget the other.
Resist the plausible-looking queen. On #84 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 #84 is rated easy
Ratings track how generous the opening is. On #84 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 45-square gap between the largest and smallest regions. Those four numbers largely explain the easy 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.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
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 #84 is row 3.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 8 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #84?
Puzzle #84 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #84 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #84, 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 #84?
Start with the orange region — the most constrained color on this board, with 1 square spanning rows 9–9 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.