12x12 Queens Puzzle #264
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 23-step reasoning is written out below.
Previous puzzle · All 12x12 puzzles · Next puzzle
Step-by-step solution for puzzle #264
What follows is a complete solve of puzzle #264, worked the way a person works it rather than the way a computer does. It runs to 23 deductions — 12 queen placements and 11 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: region shadow, region confined to one line, last square in a region, pairing argument on rows.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Wherever indigo 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 2. Blue is down to 4 candidates in rows 1–3, columns 3 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 3. The indigo region has 3 squares left, spread across rows 1 and 2, 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. All of the red region's remaining squares (2 open squares) lie in column 3. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 13 squares come off.
Step 5. Blue cannot leave column 4 — every one of its open squares (3 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 8 squares in all.
The board after step 5 — 0 of 12 queens placed. Step 6. All of the sea green 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: 12 squares come off.
Step 7. Only row 12, column 11 is still open in rose, and since each colour must take a queen, that settles it. Everything else in row 12, in column 11, in rose and in the ring around that square is now dead.
Step 8. The sea green region is down to one open square, at row 10, column 12. Every colour holds exactly one queen, so that square is forced. Row 10 and column 12 are now spoken for, along with the squares touching that queen.
Step 9. The pale yellow region has 5 squares left, spread across rows 6–9, columns 9 and 10. 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.
The board after step 9 — 2 of 12 queens placed. Step 10. All of the teal region's remaining squares (2 open squares) lie in row 11. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.
Step 11. Khaki cannot leave column 8 — 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: 11 squares in all.
Step 12. All of the pale yellow region's remaining squares (3 open squares) lie in column 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.
Step 13. Red, pale yellow, khaki, and orange are now boxed into rows 6–9 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 12 squares at once.
Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: slate has one square left, so its queen takes row 5, column 9; amber has one square left, so its queen takes row 2, column 7; indigo has one square left, so its queen takes row 1, column 1; blue has one square left, so its queen takes row 3, column 4; teal has one square left, so its queen takes row 11, column 2; mint has one square left, so its queen takes row 4, column 6; orange has one square left, so its queen takes row 9, column 5; khaki has one square left, so its queen takes row 8, column 8; pale yellow has one square left, so its queen takes row 7, column 10; red has one square left, so its queen takes row 6, column 3. 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 6, column 3.
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 #264
This is puzzle number 264 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 #264, 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 #264 are: red (2 squares), sea green (2 squares), rose (3 squares), indigo (4 squares), blue (5 squares), orange (7 squares), teal (12 squares), slate (13 squares), amber (16 squares), pale yellow (16 squares), khaki (22 squares), mint (42 squares). An even split would hand every color 12 squares, so the red region is running 10 squares under that average while the mint region is the roomiest area on the grid with 42 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Red is working with 2 squares while mint has 42 to spare, a gap of 40. 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. Red is the tidiest shape on #264, packing 100% of its 2×1 bounding box, whereas slate straggles across its 5×5 box at just 52%. 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.
Red: 2 squares across rows 6–7, columns 3–3. It sits wholly inside column 3, so its queen must be in that column.
Sea green covers 2 squares, spanning 2 rows and 1 column (rows 10–11, columns 12–12). It sits wholly inside column 12, so its queen must be in that column.
Rose occupies 3 squares within rows 11–12, columns 11–12.
Indigo: 4 squares across rows 1–2, columns 1–3.
Blue covers 5 squares, spanning 3 rows and 3 columns (rows 1–3, columns 2–4).
Orange occupies 7 squares within rows 6–9, columns 5–7.
Teal: 12 squares across rows 10–12, columns 1–4.
Slate covers 13 squares, spanning 5 rows and 5 columns (rows 3–7, columns 7–11).
Amber occupies 16 squares within rows 1–4, columns 7–12.
Pale yellow: 16 squares across rows 4–9, columns 9–12.
Khaki covers 22 squares, spanning 5 rows and 7 columns (rows 8–12, columns 5–11).
Mint occupies 42 squares within rows 1–9, columns 1–7.
Where to start on this board
The most constrained color is where a solve should begin, and on #264 that is red with 2 squares. It reaches from row 6 to row 7 and from column 3 to column 3 — 2 rows 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 red stays inside column 3, its queen belongs to that column and no other region may use it. Cross out the rest of column 3 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
Work outward to the other flat regions next. Indigo, sea green 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 #264
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 5, 10 and 12 are the narrowest, touched by only 3 colors. Columns 1, 4, 5, 6 and 8 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 7 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
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 #264 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 #264 is rated medium
Ratings track how generous the opening is. On #264 you get two regions pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 3 colors, and a 40-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.
Stuck on #264 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.
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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 12x12 Queens puzzle #264?
Puzzle #264 is rated medium. Its smallest color region holds 2 squares, 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 #264 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #264, 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 #264?
Start with the red region — the most constrained color on this board, with 2 squares spanning rows 6–7 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.