12x12 Queens Puzzle #11
Difficulty: hard. 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 24-step reasoning is written out below.
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Step-by-step solution for puzzle #11
What follows is a complete solve of puzzle #11, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 12 queen placements and 12 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: region shadow, pairing argument on rows, pairing argument on columns, last square in a region, region confined to one line, last square in a column.
Show the full walkthrough (20 steps) — contains the solution
Step 1. Wherever red ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 2. Teal is down to 3 candidates in rows 11 and 12, columns 10 and 11, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about teal itself.
Step 3. Red and pale yellow are now boxed into rows 6 and 7 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 13 squares at once.
Step 4. Here is the pairing argument. The blue, pale yellow, orange, and teal regions have no open squares outside columns 9–12 — 4 colours confined to 4 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 13 squares come off in one stroke.
Step 5. Only row 12, column 8 is still open in rose, and since each colour must take a queen, that settles it. That clears row 12, column 8, the rest of the rose region and every square adjacent to it — diagonals included.
The board after step 5 — 1 of 12 queens placed. Step 6. The khaki region is down to one open square, at row 8, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 7, the remaining khaki squares, and the ring of neighbours around row 8, column 7.
Step 7. Only row 11, column 11 is still open in teal, and since each colour must take a queen, that settles it. Everything else in row 11, in column 11, in teal and in the ring around that square is now dead.
Step 8. Indigo is down to 6 candidates in rows 1–3, columns 4–6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 9. Orange cannot leave row 9 — every one of its open squares (2 open squares) sits there. That row belongs to orange, so every square on it owned by another colour is dead: 5 squares in all.
Step 10. 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.
The board after step 10 — 4 of 12 queens placed. Step 11. Indigo is down to 3 candidates in rows 1 and 3, columns 4 and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 12. The mint region has 3 squares left, spread across rows 4 and 5, columns 4 and 5. 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. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 6 squares in all.
Step 14. Column 9 now has exactly one open square, row 7, column 9, in the pale yellow region. Columns must each hold a queen, so that placement is forced. That clears row 7, column 9, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 15. Only row 6, column 1 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 1, in red and in the ring around that square is now dead.
The board after step 15 — 6 of 12 queens placed. Step 16. Column 10 now has exactly one open square, row 9, column 10, in the orange region. Columns must each hold a queen, so that placement is forced. Everything else in row 9, in column 10, in orange and in the ring around that square is now dead.
Step 17. Amber is down to 5 candidates in rows 2–4, columns 2–4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 18. The slate region has 2 squares left, spread across rows 4 and 5, columns 2 and 3. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.
Step 19. Mint cannot leave column 5 — 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: 1 square in all.
Step 20. 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: 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 2; 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 5; 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.
The board after step 20 — 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 #11
12x12 puzzle #11 is rated hard. 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.
There is exactly one way to finish #11, 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 #11 are: red (3 squares), teal (3 squares), rose (5 squares), mint (7 squares), pale yellow (7 squares), blue (12 squares), orange (12 squares), amber (13 squares), indigo (16 squares), khaki (16 squares), sea green (20 squares), slate (30 squares). An even split would hand every color 12 squares, so the red region is running 9 squares under that average while the slate region is the roomiest area on the grid with 30 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Red has only 3 places to put its queen; slate has 30. The spread between them on this board is 27 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 mint is the most solid, filling 88% of its 2×4 box, and khaki the most diffuse at 53% of its 5×6 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 hard board.
Red occupies 3 squares within rows 6–7, columns 1–2.
Teal: 3 squares across rows 11–12, columns 10–11.
Rose covers 5 squares, spanning 4 rows and 2 columns (rows 9–12, columns 8–9).
Mint occupies 7 squares within rows 4–5, columns 4–7.
Pale yellow: 7 squares across rows 6–7, columns 9–12.
Blue covers 12 squares, spanning 5 rows and 4 columns (rows 1–5, columns 9–12).
Orange occupies 12 squares within rows 8–12, columns 10–12.
Amber: 13 squares across rows 1–4, columns 1–5.
Indigo covers 16 squares, spanning 3 rows and 7 columns (rows 1–3, columns 4–10).
Khaki occupies 16 squares within rows 4–8, columns 5–10.
Sea green: 20 squares across rows 7–12, columns 3–8.
Slate covers 30 squares, spanning 10 rows and 5 columns (rows 3–12, columns 1–5).
Where to start on this board
Begin where the choices are fewest. On this board that means red, which owns just 3 squares inside a 2×2 window running from row 6 to row 7 and column 1 to column 2. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Red spans 2 rows, so it stays open for now — but it is still the shortest list on the board at 3 squares. Work through those squares first and mark the ones that would leave another region homeless. That pruning is what makes the rest of the grid legible.
Work outward to the other flat regions next. Mint, pale yellow and teal 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 #11
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 2 are the narrowest, touched by only 3 colors. Columns 1, 2, 3 and 12 are the tightest columns, with 3 colors.
A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 11 rows. Row 12 is the crowded extreme at 5 colors — worth revisiting later, not now.
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.
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 #11 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 #11 is rated hard
The hard label comes out of four measurable things about #11: it has no region locked to one row or column, its smallest color spans 3 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 27. Boards that score tight on those measures give up their first queen fast.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a hard board from the rest.
Stuck on #11 despite the hard 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.
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 #11 the place to look for one is around row 1.
A non-spoiling distribution check: solved, #11 carries 4 queens on the perimeter against 8 queens inside, with the corners empty. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.
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 #11?
Puzzle #11 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 3 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 12x12 puzzle #11 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #11, 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 #11?
Start with the red region — the most constrained color on this board, with 3 squares spanning rows 6–7 and columns 1–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.