12x12 Queens Puzzle #166
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.
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Step-by-step solution for puzzle #166
What follows is a complete solve of puzzle #166, 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 proof by contradiction.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, line confined to one region, pairing argument on columns, last square in a row, proof by contradiction, last square in a column.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Only row 1, column 6 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 6, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The rose region is down to one open square, at row 12, column 11. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 12 and column 11, the remaining rose squares, and the ring of neighbours around row 12, column 11.
Step 3. Orange cannot leave row 9 — every one of its open squares (2 open squares) sits there. That row belongs to orange, so every square on it owned by another colour is dead: 12 squares in all.
Step 4. Wherever sea green ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
The board after step 4 — 2 of 12 queens placed. Step 5. 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 7 squares at once.
Step 6. All of the khaki region's remaining squares (2 open squares) lie in row 8. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.
Step 7. Column 12 has only red squares left (6 open squares). The column needs a queen and only red can provide one, which pins that colour to this column and kills its squares elsewhere: 8 squares.
Step 8. Here is the pairing argument. The blue, orange, and sea green regions have no open squares outside columns 1–3 — 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: 12 squares come off in one stroke.
Step 9. Only row 7, column 4 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 4, the rest of the pale yellow region and every square adjacent to it — diagonals included.
The board after step 9 — 3 of 12 queens placed. Step 10. The khaki region is down to one open square, at row 8, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 7, the remaining khaki squares, and the ring of neighbours around row 8, column 7.
Step 11. Look along row 6: one square survives, row 6, column 12, which belongs to red. A row cannot be left empty, so the queen goes there. That clears row 6, column 12, the rest of the red region and every square adjacent to it — diagonals included.
Step 12. The mint region has 6 squares left, spread across rows 3–5, columns 5, 8, 9, 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 13. Slate cannot leave column 5 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 1 square in all.
The board after step 13 — 5 of 12 queens placed. Step 14. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 8. Follow the forced moves that would create and the sea green region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 15. Test row 2, column 9 for amber: place a queen there, run the forced consequences through, and column 10 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 16. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 9. Follow the forced moves that would create and the mint region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 17. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 10; column 3 is down to one square, putting a queen on row 10, column 3; orange has one square left, so its queen takes row 9, column 1; blue has one square left, so its queen takes row 3, column 2; 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 8; teal has one square left, so its queen takes row 11, column 9. 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 9.
The board after step 17 — 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 #166
Board #166 of the 12x12 bank carries a medium rating. Its 144 squares are shared between twelve colored regions, and the solution places twelve queens: one per row, one per column, one per region, none of them side by side or corner to corner.
As with every puzzle in this collection, #166 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.
How the colors divide this board
From smallest to largest, the regions on puzzle #166 are: indigo (1 square), rose (1 square), orange (2 squares), sea green (7 squares), blue (10 squares), amber (14 squares), slate (14 squares), teal (14 squares), mint (15 squares), pale yellow (19 squares), red (22 squares), khaki (25 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the khaki region is the roomiest area on the grid with 25 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 1 place to put its queen; khaki has 25. The spread between them on this board is 24 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; mint is the loosest, occupying only 43% of its 5×7 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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 covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 6–6). It never leaves row 1, so its queen is locked to that row.
Rose occupies 1 square within rows 12–12, columns 11–11. It never leaves row 12, so its queen is locked to that row.
Orange: 2 squares across rows 9–9, columns 1–2. It never leaves row 9, so its queen is locked to that row.
Sea green covers 7 squares, spanning 3 rows and 3 columns (rows 10–12, columns 1–3).
Blue occupies 10 squares within rows 1–4, columns 1–4.
Amber: 14 squares across rows 1–4, columns 6–11.
Slate covers 14 squares, spanning 7 rows and 3 columns (rows 1–7, columns 4–6).
Teal occupies 14 squares within rows 10–12, columns 3–10.
Mint: 15 squares across rows 3–7, columns 5–11.
Pale yellow covers 19 squares, spanning 7 rows and 4 columns (rows 3–9, columns 1–4).
Red occupies 22 squares within rows 1–8, columns 8–12.
Khaki: 25 squares across rows 8–12, columns 4–12.
Where to start on this board
Indigo is the natural opening on #166: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 6–6. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.
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. Orange 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 #166
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 8, 9, 10 and 11 are the narrowest, touched by only 3 colors. Column 12 is the tightest column, with 2 colors.
A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 11 rows. Row 4 is the crowded extreme at 6 colors — worth revisiting later, not now.
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.
Keep alternating between two views of the board. In the region view you ask which of the twelve colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 12x12 board unwinds.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #166 is rated medium
Ratings track how generous the opening is. On #166 you get five regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 24-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
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.
A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.
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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #166 the place to look for one is around column 12.
A non-spoiling distribution check: solved, #166 carries 4 queens on the perimeter against 8 queens inside, with the corners empty. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.
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 #166?
Puzzle #166 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #166 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #166, 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 #166?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.