12x12 Queens Puzzle #140
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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #140
What follows is a complete solve of puzzle #140, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 12 queen placements and 7 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 confined to one line, region shadow, pairing argument on rows, line confined to one region, last square in a row, last square in a column.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Only row 3, column 1 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 1, the rest of the blue region and every square adjacent to it — diagonals included.
Step 2. The rose region is down to one open square, at row 12, column 12. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 12 and column 12, the remaining rose squares, and the ring of neighbours around row 12, column 12.
Step 3. Only row 10, column 11 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 11, in sea green and in the ring around that square is now dead.
Step 4. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 9 squares come off.
Step 5. Slate is down to 2 candidates in rows 4 and 5, columns 2 and 3, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about slate itself.
The board after step 5 — 3 of 12 queens placed. Step 6. Here is the pairing argument. The amber and mint regions have no open squares outside rows 2 and 4 — 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: 3 squares come off in one stroke.
Step 7. Only row 5, column 2 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 2, in slate and in the ring around that square is now dead.
Step 8. The teal region is down to one open square, at row 11, column 3. Every colour holds exactly one queen, so that square is forced. Row 11 and column 3 are now spoken for, along with the squares touching that queen.
Step 9. Khaki cannot leave column 4 — 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: 6 squares in all.
The board after step 9 — 5 of 12 queens placed. Step 10. The mint region is down to one open square, at row 4, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 5, the remaining mint squares, and the ring of neighbours around row 4, column 5.
Step 11. Red is down to 3 candidates in rows 6, 8, and 9, columns 6 and 10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 12. The orange region has 4 squares left, spread across rows 8 and 9, columns 7 and 9. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 13. Row 7 has only pale yellow squares left (5 open squares). The row needs a queen and only pale yellow can provide one, which pins that colour to this row and kills its squares elsewhere: 4 squares.
Step 14. 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: row 6 is down to one square, putting a queen on row 6, column 10; indigo has one square left, so its queen takes row 1, column 9; column 8 is down to one square, putting a queen on row 7, column 8; orange has one square left, so its queen takes row 9, column 7; amber has one square left, so its queen takes row 2, column 6; khaki has one square left, so its queen takes row 8, column 4. 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 4.
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 #140
This is puzzle number 140 in the 12x12 collection, rated medium. 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 #140 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 #140 are: blue (1 square), rose (1 square), indigo (4 squares), sea green (6 squares), mint (9 squares), orange (12 squares), pale yellow (14 squares), khaki (14 squares), slate (15 squares), teal (16 squares), amber (18 squares), red (34 squares). An even split would hand every color 12 squares, so the blue region is running 11 squares under that average while the red region is the roomiest area on the grid with 34 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Blue is working with 1 square while red has 34 to spare, a gap of 33. Attack the cramped end and the roomy regions tend to resolve themselves.
Square count is only half the story — how a color is arranged counts too. Blue is the tidiest shape on #140, packing 100% of its 1×1 bounding box, whereas red straggles across its 9×11 box at just 34%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Below is every color on #140 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.
Blue occupies 1 square within rows 3–3, columns 1–1. It never leaves row 3, so its queen is locked to that row.
Rose: 1 square across rows 12–12, columns 12–12. It never leaves row 12, so its queen is locked to that row.
Indigo covers 4 squares, spanning 2 rows and 3 columns (rows 1–2, columns 9–11).
Sea green occupies 6 squares within rows 9–12, columns 11–12.
Mint: 9 squares across rows 1–5, columns 3–5.
Orange covers 12 squares, spanning 5 rows and 4 columns (rows 8–12, columns 7–10).
Pale yellow occupies 14 squares within rows 6–10, columns 6–10.
Khaki: 14 squares across rows 7–11, columns 2–8.
Slate covers 15 squares, spanning 8 rows and 3 columns (rows 1–8, columns 1–3).
Teal occupies 16 squares within rows 8–12, columns 1–7.
Amber: 18 squares across rows 1–4, columns 5–10.
Red covers 34 squares, spanning 9 rows and 11 columns (rows 1–9, columns 2–12).
Where to start on this board
The most constrained color is where a solve should begin, and on #140 that is blue with 1 square. It reaches from row 3 to row 3 and from column 1 to column 1 — 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 blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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 indigo 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 #140
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 5 and 6 are the narrowest, touched by only 3 colors. Columns 1, 11 and 12 are the tightest columns, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 11 rows. Row 9 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.
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.
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 #140 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 #140 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers four single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 33. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
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 row 5 on this board.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 9 queens in the middle, including 1 in a corner. 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 #140?
Puzzle #140 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #140 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #140, 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 #140?
Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.