12x12 Queens Puzzle #197
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 25-step reasoning is written out below.
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Step-by-step solution for puzzle #197
What follows is a complete solve of puzzle #197, worked the way a person works it rather than the way a computer does. It runs to 25 deductions — 12 queen placements and 13 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 shadow, region confined to one line, pairing argument on rows, pairing argument on columns, last square in a row, proof by contradiction, last square in a column.
Show the full walkthrough (21 steps) — contains the solution
Step 1. Only row 1, column 11 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 11, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The blue region is down to one open square, at row 3, column 12. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 12, the remaining blue squares, and the ring of neighbours around row 3, column 12.
Step 3. The khaki region has 3 squares left, spread across rows 8 and 9, columns 5 and 6. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.
Step 4. Wherever orange ends up among its 3 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
Step 5. Rose cannot leave row 12 — every one of its open squares (2 open squares) sits there. That row belongs to rose, so every square on it owned by another colour is dead: 10 squares in all.
The board after step 5 — 2 of 12 queens placed. Step 6. The teal region has 5 squares left, spread across rows 10 and 11, columns 6, 7, 8, and 10. 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. Khaki and orange are now boxed into rows 8 and 9 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 10 squares at once.
Step 8. Pale yellow is down to 3 candidates in rows 7 and 10, columns 2, 4, and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 9. Sea green and teal are now boxed into rows 10 and 11 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 6 squares at once.
Step 10. The pale yellow region is down to one open square, at row 7, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 2, the remaining pale yellow squares, and the ring of neighbours around row 7, column 2.
Step 11. Only row 12, column 3 is still open in rose, and since each colour must take a queen, that settles it. Everything else in row 12, in column 3, in rose and in the ring around that square is now dead.
The board after step 11 — 4 of 12 queens placed. Step 12. The slate region has 3 squares left, spread across rows 5 and 6, columns 4 and 5. 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. Wherever red 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 14. Here is the pairing argument. The amber, slate, and sea green regions have no open squares outside columns 1, 4, and 5 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 4 squares come off in one stroke.
Step 15. Only row 8, column 6 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 6, in khaki and in the ring around that square is now dead.
Step 16. The orange region is down to one open square, at row 9, column 8. Every colour holds exactly one queen, so that square is forced. Row 9 and column 8 are now spoken for, along with the squares touching that queen.
The board after step 16 — 6 of 12 queens placed. Step 17. Look along row 10: one square survives, row 10, column 1, which belongs to sea green. A row cannot be left empty, so the queen goes there. Everything else in row 10, in column 1, in sea green and in the ring around that square is now dead.
Step 18. All of the amber 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: 1 square come off.
Step 19. Wherever red 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 20. Now a short proof by contradiction. Suppose the amber queen went to row 4, column 4. Follow the forced moves that would create and the red region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 21. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 4; column 9 is down to one square, putting a queen on row 6, column 9; slate has one square left, so its queen takes row 5, column 5; mint has one square left, so its queen takes row 4, column 7; teal has one square left, so its queen takes row 11, column 10. 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 10.
The board after step 21 — 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 #197
Puzzle #197 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 #197, 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 #197 are: indigo (1 square), rose (2 squares), khaki (3 squares), orange (3 squares), blue (6 squares), pale yellow (8 squares), slate (9 squares), sea green (16 squares), amber (20 squares), teal (20 squares), mint (27 squares), red (29 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the red region is the roomiest area on the grid with 29 squares.
Every color takes one queen and one only, whatever its size — which is why the 28 square difference between indigo and red 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.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #197, packing 100% of its 1×1 bounding box, whereas pale yellow straggles across its 4×5 box at just 40%. 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.
Indigo occupies 1 square within rows 1–1, columns 11–11. It never leaves row 1, so its queen is locked to that row.
Rose: 2 squares across rows 12–12, columns 2–3. It never leaves row 12, so its queen is locked to that row.
Khaki covers 3 squares, spanning 2 rows and 2 columns (rows 8–9, columns 5–6).
Orange occupies 3 squares within rows 8–9, columns 7–8.
Blue: 6 squares across rows 1–3, columns 10–12.
Pale yellow covers 8 squares, spanning 4 rows and 5 columns (rows 7–10, columns 1–5).
Slate occupies 9 squares within rows 5–7, columns 1–5.
Sea green: 16 squares across rows 9–12, columns 1–6.
Amber covers 20 squares, spanning 5 rows and 4 columns (rows 1–5, columns 1–4).
Teal occupies 20 squares within rows 8–12, columns 6–12.
Mint: 27 squares across rows 1–6, columns 5–10.
Red covers 29 squares, spanning 9 rows and 8 columns (rows 3–11, columns 5–12).
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 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 #197.
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. Rose, khaki and orange 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 #197
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 4, 6, 7, 11 and 12 are the narrowest, touched by only 3 colors. Columns 9 and 12 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 9 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.
The whole middle game reduces to two sweeps. Sweep the twelve colors looking for one with a single square left; sweep the twelve rows and twelve columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 12x12 solve from a stalled one.
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 #197 is rated easy
Ratings track how generous the opening is. On #197 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 28-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
Stuck on #197 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.
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 #197 is row 2.
As a sanity check rather than a spoiler: the finished solution to #197 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 #197?
Puzzle #197 is rated easy. 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 #197 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #197, 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 #197?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.