12x12 Queens Puzzle #141
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 18-step reasoning is written out below.
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Step-by-step solution for puzzle #141
What follows is a complete solve of puzzle #141, 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 pairing argument on columns.
Techniques this board needs: last square in a region, region confined to one line, pairing argument on rows, region shadow, pairing argument on columns, last square in a row.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Only row 1, column 2 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 2, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The blue region is down to one open square, at row 3, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 1, the remaining blue squares, and the ring of neighbours around row 3, column 1.
Step 3. Amber cannot leave row 2 — every one of its open squares (2 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 7 squares in all.
Step 4. All of the pale yellow 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: 12 squares come off.
Step 5. Only row 2, column 12 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 12, the rest of the amber region and every square adjacent to it — diagonals included.
The board after step 5 — 3 of 12 queens placed. Step 6. Here is the pairing argument. The orange, sea green, teal, and rose regions have no open squares outside rows 9–12 — 4 colours confined to 4 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 9 squares come off in one stroke.
Step 7. Only row 8, column 7 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 7, in khaki and in the ring around that square is now dead.
Step 8. The pale yellow region is down to one open square, at row 7, column 9. Every colour holds exactly one queen, so that square is forced. Row 7 and column 9 are now spoken for, along with the squares touching that queen.
Step 9. The red region has 3 squares left, spread across rows 5 and 6, columns 10 and 11. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.
The board after step 9 — 5 of 12 queens placed. Step 10. Here is the pairing argument. The red and sea green regions have no open squares outside columns 10 and 11 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 2 squares come off in one stroke.
Step 11. Look along row 9: one square survives, row 9, column 4, which belongs to orange. A row cannot be left empty, so the queen goes there. That clears row 9, column 4, the rest of the orange region and every square adjacent to it — diagonals included.
Step 12. Row 10 has a single open square left, at column 10 in the sea green region. Every row needs a queen, so this one is forced. Cross out the whole of row 10 and column 10, the remaining sea green squares, and the ring of neighbours around row 10, column 10.
Step 13. Teal cannot leave column 8 — 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: 2 squares in all.
Step 14. 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: slate has one square left, so its queen takes row 5, column 6; red has one square left, so its queen takes row 6, column 11; mint has one square left, so its queen takes row 4, column 3; rose has one square left, so its queen takes row 12, column 5; teal has one square left, so its queen takes row 11, column 8. 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 8.
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 #141
Puzzle #141 is one of the easy-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.
There is exactly one way to finish #141, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these twelve regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #141 are: indigo (1 square), pale yellow (2 squares), blue (4 squares), rose (6 squares), amber (8 squares), khaki (9 squares), sea green (9 squares), teal (9 squares), orange (14 squares), red (15 squares), slate (24 squares), mint (43 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the mint region is the roomiest area on the grid with 43 squares.
Every color takes one queen and one only, whatever its size — which is why the 42 square difference between indigo and mint 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. Indigo is the tidiest shape on #141, packing 100% of its 1×1 bounding box, whereas slate straggles across its 7×8 box at just 43%. 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.
Indigo: 1 square across rows 1–1, columns 2–2. It never leaves row 1, so its queen is locked to that row.
Pale yellow covers 2 squares, spanning 2 rows and 1 column (rows 7–8, columns 9–9). It sits wholly inside column 9, so its queen must be in that column.
Blue occupies 4 squares within rows 1–3, columns 1–2.
Rose: 6 squares across rows 11–12, columns 3–7.
Amber covers 8 squares, spanning 3 rows and 5 columns (rows 1–3, columns 8–12).
Khaki occupies 9 squares within rows 7–10, columns 5–8.
Sea green: 9 squares across rows 9–12, columns 10–12.
Teal covers 9 squares, spanning 4 rows and 4 columns (rows 9–12, columns 7–10).
Orange occupies 14 squares within rows 9–12, columns 1–5.
Red: 15 squares across rows 4–10, columns 10–12.
Slate covers 24 squares, spanning 7 rows and 8 columns (rows 2–8, columns 5–12).
Mint occupies 43 squares within rows 1–9, columns 1–7.
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 2 and 2. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #141.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Pale yellow 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 #141
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 4, 5 and 6 are the narrowest, touched by only 3 colors. Columns 1, 3 and 4 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.
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 #141 is rated easy
Ratings track how generous the opening is. On #141 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 42-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
Stuck on #141 despite the easy rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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 #141 is row 4.
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 #141?
Puzzle #141 is rated easy. 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 #141 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #141, 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 #141?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.