12x12 Queens Puzzle #232
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 24-step reasoning is written out below.
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Step-by-step solution for puzzle #232
What follows is a complete solve of puzzle #232, 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 proof by contradiction.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, pairing argument on columns, proof by contradiction, last square in a row.
Show the full walkthrough (19 steps) — contains the solution
Step 1. Only row 5, column 1 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 1, the rest of the slate region and every square adjacent to it — diagonals included.
Step 2. The red region is down to one open square, at row 6, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 4, the remaining red squares, and the ring of neighbours around row 6, column 4.
Step 3. 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: 10 squares in all.
Step 4. The amber region is down to one open square, at row 2, column 6. Every colour holds exactly one queen, so that square is forced. Row 2 and column 6 are now spoken for, along with the squares touching that queen.
Step 5. Blue is down to 3 candidates in rows 3 and 4, columns 10 and 12, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about blue itself.
The board after step 5 — 3 of 12 queens placed. Step 6. The khaki region has 6 squares left, spread across rows 7, 8, 10, and 12, columns 2 and 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. Wherever orange ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 8. All of the rose region's remaining squares (2 open squares) lie in row 12. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 7 squares come off.
Step 9. Blue and mint are now boxed into rows 3 and 4 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 5 squares at once.
Step 10. Here is the pairing argument. The mint, khaki, sea green, and rose regions have no open squares outside columns 2, 3, 5, and 7 — 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: 7 squares come off in one stroke.
The board after step 10 — 3 of 12 queens placed. Step 11. Test row 1, column 10 for indigo: place a queen there, run the forced consequences through, and the rose region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 12. The indigo region is down to one open square, at row 1, column 9. Every colour holds exactly one queen, so that square is forced. Row 1 and column 9 are now spoken for, along with the squares touching that queen.
Step 13. Wherever pale yellow ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 14. All of the orange region's remaining squares (3 open squares) lie in column 12. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
The board after step 14 — 4 of 12 queens placed. Step 15. Only row 3, column 10 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 10, in blue and in the ring around that square is now dead.
Step 16. The pale yellow region is down to one open square, at row 7, column 8. Every colour holds exactly one queen, so that square is forced. Row 7 and column 8 are now spoken for, along with the squares touching that queen.
Step 17. Khaki is down to 4 candidates in rows 8 and 10, columns 2 and 3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 18. All of the orange 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: 1 square come off.
Step 19. 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 9 is down to one square, putting a queen on row 9, column 12; teal has one square left, so its queen takes row 11, column 11; sea green has one square left, so its queen takes row 10, column 5; rose has one square left, so its queen takes row 12, column 7; mint has one square left, so its queen takes row 4, column 3; khaki has one square left, so its queen takes row 8, 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 8, column 2.
The board after step 19 — 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 #232
This is puzzle number 232 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.
There is exactly one way to finish #232, 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 #232 are: slate (1 square), indigo (2 squares), red (3 squares), orange (5 squares), amber (6 squares), rose (6 squares), sea green (12 squares), blue (13 squares), khaki (17 squares), teal (18 squares), mint (22 squares), pale yellow (39 squares). An even split would hand every color 12 squares, so the slate region is running 11 squares under that average while the pale yellow region is the roomiest area on the grid with 39 squares.
Every color takes one queen and one only, whatever its size — which is why the 38 square difference between slate and pale yellow 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 indigo region is the most compact color here, filling 100% of the 1×2 box that encloses it, while rose is the most scattered at 50% of its 4×3 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
Below is every color on #232 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.
Slate covers 1 square, spanning 1 row and 1 column (rows 5–5, columns 1–1). It never leaves row 5, so its queen is locked to that row.
Indigo occupies 2 squares within rows 1–1, columns 9–10. It never leaves row 1, so its queen is locked to that row.
Red: 3 squares across rows 5–6, columns 3–4.
Orange covers 5 squares, spanning 4 rows and 2 columns (rows 6–9, columns 11–12).
Amber occupies 6 squares within rows 1–2, columns 3–7.
Rose: 6 squares across rows 9–12, columns 5–7.
Sea green covers 12 squares, spanning 5 rows and 4 columns (rows 8–12, columns 2–5).
Blue occupies 13 squares within rows 1–6, columns 9–12.
Khaki: 17 squares across rows 6–12, columns 1–4.
Teal covers 18 squares, spanning 5 rows and 5 columns (rows 8–12, columns 8–12).
Mint occupies 22 squares within rows 1–5, columns 1–7.
Pale yellow: 39 squares across rows 1–11, columns 5–11.
Where to start on this board
The most constrained color is where a solve should begin, and on #232 that is slate with 1 square. It reaches from row 5 to row 5 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 slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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. Indigo, amber and red 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 #232
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3, 4 and 7 are the narrowest, touched by only 3 colors. Column 8 is the tightest column, with 2 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 9 is the crowded extreme at 6 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.
A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.
Why puzzle #232 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers three single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 38. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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 medium board from the rest.
Stuck on #232 despite the medium 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 #232 is column 8.
A non-spoiling distribution check: solved, #232 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 #232?
Puzzle #232 is rated medium. 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 #232 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #232, 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 #232?
Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.