12x12 Queens Puzzle #35
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 32-step reasoning is written out below.
Previous puzzle · All 12x12 puzzles · Next puzzle
Step-by-step solution for puzzle #35
What follows is a complete solve of puzzle #35, worked the way a person works it rather than the way a computer does. It runs to 32 deductions — 12 queen placements and 20 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 proof by contradiction.
Techniques this board needs: last square in a region, region shadow, pairing argument on rows, region confined to one line, pairing argument on columns, proof by contradiction, last square in a row.
Show the full walkthrough (27 steps) — contains the solution
Step 1. Only row 2, column 7 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 7, the rest of the amber region and every square adjacent to it — diagonals included.
Step 2. Red is down to 5 candidates in rows 6–9, columns 11 and 12, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 3. The rose region has 4 squares left, spread across rows 10–12, columns 1 and 2. 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.
Step 4. Here is the pairing argument. The sea green, teal, and rose regions have no open squares outside rows 10–12 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 13 squares come off in one stroke.
Step 5. Orange is down to 4 candidates in rows 8 and 9, columns 4–6, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about orange itself.
Step 6. The khaki region has 4 squares left, spread across rows 8 and 9, columns 1–3. 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.
Step 7. Khaki and orange are now boxed into rows 8 and 9 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 11 squares at once.
The board after step 7 — 1 of 12 queens placed. Step 8. All of the red region's remaining squares (2 open squares) lie in column 12. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 8 squares come off.
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: 7 squares in all.
Step 10. The red region is down to one open square, at row 6, column 12. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 12, the remaining red squares, and the ring of neighbours around row 6, column 12.
Step 11. Slate is down to 3 candidates in rows 4 and 5, columns 1 and 3, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about slate itself.
Step 12. Here is the pairing argument. The slate, khaki, and rose regions have no open squares outside columns 1–3 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 7 squares come off in one stroke.
Step 13. Only row 1, column 4 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 4, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 14. Mint is down to 5 candidates in rows 3–5, columns 8–11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.
The board after step 14 — 3 of 12 queens placed. Step 15. Test row 3, column 10 for blue: place a queen there, run the forced consequences through, and column 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 16. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 11. Follow the forced moves that would create and column 6 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 17. Test row 4, column 5 for blue: place a queen there, run the forced consequences through, and the sea green region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 18. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 6. Follow the forced moves that would create and the sea green region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 19. Test row 4, column 8 for blue: place a queen there, run the forced consequences through, and row 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 20. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 10. Follow the forced moves that would create and row 3 runs out of open squares entirely — impossible. So that square is ruled out for good.
The board after step 20 — 3 of 12 queens placed. Step 21. Blue and orange are now boxed into columns 5 and 6 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 2 squares at once.
Step 22. Here is the pairing argument. The pale yellow and sea green regions have no open squares outside columns 8 and 9 — 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: 3 squares come off in one stroke.
Step 23. 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 3 is down to one square, putting a queen on row 3, column 5; orange has one square left, so its queen takes row 9, column 6; khaki has one square left, so its queen takes row 8, column 1; slate has one square left, so its queen takes row 5, column 3; mint has one square left, so its queen takes row 4, column 11; rose has one square left, so its queen takes row 12, column 2. 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 12, column 2.
Step 24. Sea green is down to 3 candidates in rows 10 and 11, columns 8 and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about sea green itself.
Step 25. The teal region is down to one open square, at row 11, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 10, the remaining teal squares, and the ring of neighbours around row 11, column 10.
Step 26. Only row 10, column 8 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 8, in sea green and in the ring around that square is now dead.
Step 27. 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.
The board after step 27 — 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 #35
12x12 puzzle #35 is rated easy. 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, #35 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 #35 are: amber (1 square), rose (4 squares), red (5 squares), khaki (6 squares), sea green (7 squares), teal (9 squares), indigo (14 squares), pale yellow (14 squares), slate (16 squares), orange (21 squares), blue (22 squares), mint (25 squares). An even split would hand every color 12 squares, so the amber region is running 11 squares under that average while the mint region is the roomiest area on the grid with 25 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Amber has only 1 place to put its queen; mint has 25. The spread between them on this board is 24 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
A region's outline can constrain it more than its size does. Here amber is the most solid, filling 100% of its 1×1 box, and pale yellow the most diffuse at 47% of its 5×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
Amber occupies 1 square within rows 2–2, columns 7–7. It never leaves row 2, so its queen is locked to that row.
Rose: 4 squares across rows 10–12, columns 1–2.
Red covers 5 squares, spanning 4 rows and 2 columns (rows 6–9, columns 11–12).
Khaki occupies 6 squares within rows 8–10, columns 1–4.
Sea green: 7 squares across rows 10–12, columns 6–9.
Teal covers 9 squares, spanning 3 rows and 3 columns (rows 10–12, columns 10–12).
Indigo occupies 14 squares within rows 1–5, columns 1–4.
Pale yellow: 14 squares across rows 5–9, columns 6–11.
Slate covers 16 squares, spanning 5 rows and 6 columns (rows 4–8, columns 1–6).
Orange occupies 21 squares within rows 8–12, columns 2–8.
Blue: 22 squares across rows 1–5, columns 4–11.
Mint covers 25 squares, spanning 7 rows and 5 columns (rows 1–7, columns 8–12).
Where to start on this board
Begin where the choices are fewest. On this board that means amber, which owns just 1 square inside a 1×1 window running from row 2 to row 2 and column 7 to column 7. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because amber never leaves row 2, its queen must be in that row — and that instantly settles row 2 for everyone else. Sweep along it and X out every square that is not amber. 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.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: red and rose span only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #35
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 3 are the narrowest, touched by only 3 colors. Columns 5 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 10 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
The middle of #35 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.
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 #35 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 #35 is rated easy
Ratings track how generous the opening is. On #35 you get two regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 24-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.
If #35 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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #35 the place to look for one is around row 1.
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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 12x12 Queens puzzle #35?
Puzzle #35 is rated easy. 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 #35 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #35, 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 #35?
Start with the amber region — the most constrained color on this board, with 1 square spanning rows 2–2 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.