12x12 Queens Puzzle #21
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 19-step reasoning is written out below.
Previous puzzle · All 12x12 puzzles · Next puzzle
Step-by-step solution for puzzle #21
What follows is a complete solve of puzzle #21, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 12 queen placements and 7 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: last square in a region, region shadow, pairing argument on columns, region confined to one line, last square in a row.
Show the full walkthrough (19 steps) — contains the solution
Step 1. Only row 4, column 11 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 11, the rest of the mint region and every square adjacent to it — diagonals included.
Step 2. 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 3. Only row 12, column 9 is still open in rose, and since each colour must take a queen, that settles it. Everything else in row 12, in column 9, in rose and in the ring around that square is now dead.
Step 4. Wherever indigo ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Amber is down to 3 candidates in rows 2 and 3, columns 2 and 3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
The board after step 5 — 3 of 12 queens placed. Step 6. The orange region has 6 squares left, spread across rows 7–11, columns 4 and 5. 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. Indigo, amber, pale yellow, and teal are now boxed into columns 1–4 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 7 squares at once.
Step 8. All of the orange region's remaining squares (4 open squares) lie in column 5. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
Step 9. Only row 3, column 6 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 6, the rest of the blue region and every square adjacent to it — diagonals included.
Step 10. The amber region is down to one open square, at row 2, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 3, the remaining amber squares, and the ring of neighbours around row 2, column 3.
The board after step 10 — 5 of 12 queens placed. Step 11. Only row 1, column 1 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 1, in indigo and in the ring around that square is now dead.
Step 12. The slate region has 4 squares left, spread across rows 5, 7, 8, and 9, columns 7 and 8. 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. Look along row 5: one square survives, row 5, column 7, which belongs to slate. A row cannot be left empty, so the queen goes there. Everything else in row 5, in column 7, in slate and in the ring around that square is now dead.
Step 14. The sea green region is down to one open square, at row 10, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 8, the remaining sea green squares, and the ring of neighbours around row 10, column 8.
The board after step 14 — 8 of 12 queens placed. Step 15. Look along row 11: one square survives, row 11, column 2, which belongs to teal. A row cannot be left empty, so the queen goes there. That clears row 11, column 2, the rest of the teal region and every square adjacent to it — diagonals included.
Step 16. All of the pale yellow region's remaining squares (2 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 17. Only row 9, column 5 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 5, the rest of the orange region and every square adjacent to it — diagonals included.
Step 18. The pale yellow region is down to one open square, at row 7, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 4, the remaining pale yellow squares, and the ring of neighbours around row 7, column 4.
Step 19. Only row 8, column 10 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 10, in khaki and in the ring around that square is now dead.
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 #21
Puzzle #21 is one of the medium-rated boards in the 12x12 set. Twelve colors divide its 144 squares, and you win by placing twelve queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
Puzzle #21 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #21 are: mint (1 square), rose (1 square), amber (4 squares), red (5 squares), indigo (6 squares), orange (7 squares), pale yellow (12 squares), sea green (13 squares), blue (14 squares), teal (19 squares), khaki (28 squares), slate (34 squares). An even split would hand every color 12 squares, so the mint region is running 11 squares under that average while the slate region is the roomiest area on the grid with 34 squares.
Every color takes one queen and one only, whatever its size — which is why the 33 square difference between mint and slate 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.
Look at arrangement as well as area. Amber is the densest region on this grid at 100% of a 2×2 frame; slate is the loosest, occupying only 47% of its 9×8 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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 medium board.
Mint: 1 square across rows 4–4, columns 11–11. It never leaves row 4, so its queen is locked to that row.
Rose covers 1 square, spanning 1 row and 1 column (rows 12–12, columns 9–9). It never leaves row 12, so its queen is locked to that row.
Amber occupies 4 squares within rows 2–3, columns 2–3.
Red: 5 squares across rows 4–6, columns 10–12.
Indigo covers 6 squares, spanning 4 rows and 3 columns (rows 1–4, columns 1–3).
Orange occupies 7 squares within rows 6–11, columns 4–5.
Pale yellow: 12 squares across rows 5–9, columns 1–4.
Sea green covers 13 squares, spanning 6 rows and 3 columns (rows 7–12, columns 6–8).
Blue occupies 14 squares within rows 1–5, columns 2–6.
Teal: 19 squares across rows 8–12, columns 1–7.
Khaki covers 28 squares, spanning 8 rows and 5 columns (rows 5–12, columns 8–12).
Slate occupies 34 squares within rows 1–9, columns 5–12.
Where to start on this board
Open on the tightest color. Here that is mint, confined to 1 square between rows 4 and 4 and columns 11 and 11. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #21.
Because mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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. Rose and amber 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 #21
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 3 colors. Columns 1, 7, 8, 9, 10 and 12 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 9, meeting 6 colors, is the opposite case — the busiest line on #21 and the last one you should expect to crack.
Working through the middle game
The middle of #21 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.
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 #21 is rated medium
The medium label comes out of four measurable things about #21: four 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 33. 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 medium board from the rest.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
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 #21 is row 1.
Here is a shape check that gives nothing away: when #21 is solved, 3 queens sit on the outer edge of the grid and 9 sit inside it, with 1 queen in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 #21?
Puzzle #21 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #21 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #21, 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 #21?
Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.