12x12 Queens Puzzle #221
Difficulty: easy. 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 21-step reasoning is written out below.
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Step-by-step solution for puzzle #221
What follows is a complete solve of puzzle #221, worked the way a person works it rather than the way a computer does. It runs to 21 deductions — 12 queen placements and 9 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, region confined to one line, pairing argument on columns, pairing argument on rows, last square in a row.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Only row 1, column 12 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 12, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. Amber is down to 3 candidates in rows 2–4, columns 9 and 11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 3. Mint cannot leave row 4 — every one of its open squares (3 open squares) sits there. That row belongs to mint, so every square on it owned by another colour is dead: 10 squares in all.
Step 4. Wherever amber ends up among its 2 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
Step 5. Teal cannot leave row 11 — every one of its open squares (2 open squares) sits there. That row belongs to teal, so every square on it owned by another colour is dead: 13 squares in all.
The board after step 5 — 1 of 12 queens placed. Step 6. The rose region is down to one open square, at row 12, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 12 and column 7, the remaining rose squares, and the ring of neighbours around row 12, column 7.
Step 7. Only row 11, column 5 is still open in teal, and since each colour must take a queen, that settles it. Everything else in row 11, in column 5, in teal and in the ring around that square is now dead.
Step 8. All of the mint region's remaining squares (2 open squares) lie in row 4. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 9. The khaki region has 4 squares left, spread across rows 8–10, columns 3 and 4. 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 — 3 of 12 queens placed. Step 10. Wherever orange ends up among its 2 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. Mint and khaki are now boxed into columns 3 and 4 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 5 squares at once.
Step 12. The orange region is down to one open square, at row 9, column 6. Every colour holds exactly one queen, so that square is forced. Row 9 and column 6 are now spoken for, along with the squares touching that queen.
Step 13. Pale yellow, khaki, and sea green are now boxed into rows 7, 8, and 10 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 7 squares at once.
Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: red has one square left, so its queen takes row 6, column 11; amber has one square left, so its queen takes row 2, column 9; slate has one square left, so its queen takes row 5, column 8; sea green has one square left, so its queen takes row 10, column 10; row 7 is down to one square, putting a queen on row 7, column 2; blue has one square left, so its queen takes row 3, column 1; khaki has one square left, so its queen takes row 8, column 4; mint has one square left, so its queen takes row 4, 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 4, 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 #221
Puzzle #221 is one of the easy-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.
There is exactly one way to finish #221, 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 #221 are: indigo (1 square), teal (2 squares), mint (3 squares), rose (3 squares), amber (10 squares), khaki (10 squares), orange (12 squares), red (14 squares), pale yellow (17 squares), blue (18 squares), sea green (20 squares), slate (34 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the slate region is the roomiest area on the grid with 34 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So indigo, with 1 square, is under far more pressure than slate with 34. A 33-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; amber is the loosest, occupying only 42% of its 4×6 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 easy board.
Indigo occupies 1 square within rows 1–1, columns 12–12. It never leaves row 1, so its queen is locked to that row.
Teal: 2 squares across rows 11–11, columns 5–6. It never leaves row 11, so its queen is locked to that row.
Mint covers 3 squares, spanning 1 row and 3 columns (rows 4–4, columns 3–5). It never leaves row 4, so its queen is locked to that row.
Rose occupies 3 squares within rows 11–12, columns 6–7.
Amber: 10 squares across rows 1–4, columns 6–11.
Khaki covers 10 squares, spanning 5 rows and 3 columns (rows 8–12, columns 3–5).
Orange occupies 12 squares within rows 7–10, columns 4–7.
Red: 14 squares across rows 2–9, columns 10–12.
Pale yellow covers 17 squares, spanning 8 rows and 5 columns (rows 5–12, columns 1–5).
Blue occupies 18 squares within rows 1–7, columns 1–4.
Sea green: 20 squares across rows 7–12, columns 8–12.
Slate covers 34 squares, spanning 8 rows and 8 columns (rows 1–8, columns 4–11).
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 12 and 12. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #221.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Mint, teal 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 #221
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3, 5, 6, 10 and 12 are the narrowest, touched by only 4 colors. Columns 1 and 2 are the tightest columns, with 2 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 4 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 8, meeting 6 colors, is the opposite case — the busiest line on #221 and the last one you should expect to crack.
Working through the middle game
The middle of #221 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.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #221 is rated easy
The easy label comes out of four measurable things about #221: four of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 4 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 easy board from the rest.
Stuck on #221 despite the easy 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.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #221, start hunting for it near column 1.
Here is a shape check that gives nothing away: when #221 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.
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Questions about this puzzle
How hard is 12x12 Queens puzzle #221?
Puzzle #221 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 4 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #221 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #221, 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 #221?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.