12x12 Queens Puzzle #268
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 27-step reasoning is written out below.
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Step-by-step solution for puzzle #268
What follows is a complete solve of puzzle #268, worked the way a person works it rather than the way a computer does. It runs to 27 deductions — 12 queen placements and 15 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, proof by contradiction.
Show the full walkthrough (20 steps) — contains the solution
Step 1. The board opens by cascading. Each of these placements is forced the moment the one before it is marked, so the whole run is really a single sweep repeated: sea green has one square left, so its queen takes row 10, column 11; slate has one square left, so its queen takes row 5, column 12; indigo has one square left, so its queen takes row 1, column 10; khaki has one square left, so its queen takes row 8, column 9; rose has one square left, so its queen takes row 12, 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 12, column 8.
Step 2. All of the amber region's remaining squares (2 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 7 squares come off.
Step 3. Only row 3, column 1 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 1, in blue and in the ring around that square is now dead.
Step 4. Pale yellow is down to 3 candidates in rows 7 and 9, columns 2–4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 5. Test row 4, column 4 for mint: place a queen there, run the forced consequences through, and the teal region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 5 — 6 of 12 queens placed. Step 6. Now a short proof by contradiction. Suppose the mint queen went to row 4, column 6. Follow the forced moves that would create and the orange region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 7. Test row 4, column 7 for mint: place a queen there, run the forced consequences through, and the orange region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 8. Now a short proof by contradiction. Suppose the mint queen went to row 6, column 4. Follow the forced moves that would create and the teal region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 9. Test row 6, column 5 for mint: place a queen there, run the forced consequences through, and the teal region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 10. Now a short proof by contradiction. Suppose the mint queen went to row 6, column 7. Follow the forced moves that would create and the teal region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
The board after step 10 — 6 of 12 queens placed. Step 11. 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: 2 squares in all.
Step 12. Wherever teal 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 13. Test row 7, column 5 for mint: place a queen there, run the forced consequences through, and row 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 14. The mint region is down to one open square, at row 4, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 5, the remaining mint squares, and the ring of neighbours around row 4, column 5.
Step 15. Test row 6, column 3 for red: place a queen there, run the forced consequences through, and the teal region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 15 — 7 of 12 queens placed. Step 16. Now a short proof by contradiction. Suppose the red queen went to row 7, column 2. Follow the forced moves that would create and the teal region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 17. Test row 7, column 3 for red: place a queen there, run the forced consequences through, and the teal region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 18. The red region is down to one open square, at row 6, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 2, the remaining red squares, and the ring of neighbours around row 6, column 2.
Step 19. Teal is down to 2 candidates in rows 9 and 11, columns 3 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about teal itself.
Step 20. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: pale yellow has one square left, so its queen takes row 7, column 4; orange has one square left, so its queen takes row 9, column 7; amber has one square left, so its queen takes row 2, column 6; teal has one square left, so its queen takes row 11, 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 11, column 3.
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 #268
This is puzzle number 268 in the 12x12 collection, rated hard. 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 #268, 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 #268 are: sea green (1 square), slate (2 squares), pale yellow (5 squares), blue (7 squares), indigo (9 squares), teal (11 squares), rose (11 squares), amber (12 squares), khaki (15 squares), red (18 squares), mint (25 squares), orange (28 squares). An even split would hand every color 12 squares, so the sea green region is running 11 squares under that average while the orange region is the roomiest area on the grid with 28 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Sea green is working with 1 square while orange has 28 to spare, a gap of 27. Attack the cramped end and the roomy regions tend to resolve themselves.
Shape matters as much as size. The slate region is the most compact color here, filling 100% of the 1×2 box that encloses it, while teal is the most scattered at 46% of its 4×6 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
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.
Sea green covers 1 square, spanning 1 row and 1 column (rows 10–10, columns 11–11). It never leaves row 10, so its queen is locked to that row.
Slate occupies 2 squares within rows 5–5, columns 11–12. It never leaves row 5, so its queen is locked to that row.
Pale yellow: 5 squares across rows 7–9, columns 2–4.
Blue covers 7 squares, spanning 3 rows and 3 columns (rows 1–3, columns 1–3).
Indigo occupies 9 squares within rows 1–4, columns 10–12.
Teal: 11 squares across rows 9–12, columns 1–6.
Rose covers 11 squares, spanning 3 rows and 5 columns (rows 10–12, columns 8–12).
Amber occupies 12 squares within rows 1–3, columns 4–10.
Khaki: 15 squares across rows 5–10, columns 9–12.
Red covers 18 squares, spanning 8 rows and 3 columns (rows 3–10, columns 1–3).
Mint occupies 25 squares within rows 2–7, columns 3–10.
Orange: 28 squares across rows 5–12, columns 4–9.
Where to start on this board
The most constrained color is where a solve should begin, and on #268 that is sea green with 1 square. It reaches from row 10 to row 10 and from column 11 to column 11 — 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 sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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. Slate is no more than two rows tall, so its queen is 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 #268
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 4, 11 and 12 are the narrowest, touched by only 3 colors. Columns 1 and 7 are the tightest columns, with 3 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 11 rows. By contrast row 10 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
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 #268 is rated hard
The hard label comes out of four measurable things about #268: three of its regions are locked to one row or column, its smallest color spans 1 square, 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.
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 #268 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.
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 1 on this board.
As a sanity check rather than a spoiler: the finished solution to #268 puts 4 queens on the outer border and 8 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 #268?
Puzzle #268 is rated hard. 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 #268 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #268, 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 #268?
Start with the sea green region — the most constrained color on this board, with 1 square spanning rows 10–10 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.