12x12 Queens Puzzle #186
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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #186
What follows is a complete solve of puzzle #186, 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 rows, pairing argument on columns, region confined to one line, last square in a row.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Only row 2, column 12 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 12, the rest of the amber region and every square adjacent to it — diagonals included.
Step 2. The mint region is down to one open square, at row 4, column 11. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 11, the remaining mint squares, and the ring of neighbours around row 4, column 11.
Step 3. The orange region has 5 squares left, spread across rows 7–10, columns 1 and 2. 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 4. Wherever teal 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 5. Indigo and blue are now boxed into rows 1 and 3 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 5 squares at once.
The board after step 5 — 2 of 12 queens placed. Step 6. Here is the pairing argument. The slate and red regions have no open squares outside rows 5 and 6 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 9 squares come off in one stroke.
Step 7. Orange and teal are now boxed into columns 1 and 2 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 7 squares at once.
Step 8. 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: slate has one square left, so its queen takes row 5, column 3; red has one square left, so its queen takes row 6, column 9; indigo has one square left, so its queen takes row 1, column 10; blue has one square left, so its queen takes row 3, column 5. 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 3, column 5.
Step 9. All of the pale yellow region's remaining squares (2 open squares) lie in row 7. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 10. Only row 8, column 7 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 7, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 11. 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.
The board after step 11 — 8 of 12 queens placed. Step 12. Look along row 9: one square survives, row 9, column 2, which belongs to orange. A row cannot be left empty, so the queen goes there. That clears row 9, column 2, the rest of the orange region and every square adjacent to it — diagonals included.
Step 13. Wherever sea green ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 14. Look along row 10: one square survives, row 10, column 6, which belongs to sea green. A row cannot be left empty, so the queen goes there. Everything else in row 10, in column 6, in sea green and in the ring around that square is now dead.
Step 15. The rose region is down to one open square, at row 12, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 12 and column 8, the remaining rose squares, and the ring of neighbours around row 12, column 8.
Step 16. Only row 11, column 1 is still open in teal, and since each colour must take a queen, that settles it. Everything else in row 11, in column 1, in teal and in the ring around that square is now dead.
The board after step 16 — 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 #186
Board #186 of the 12x12 bank carries a hard 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, #186 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 #186 are: amber (1 square), orange (5 squares), teal (5 squares), mint (7 squares), indigo (10 squares), slate (10 squares), pale yellow (15 squares), red (16 squares), blue (17 squares), rose (17 squares), khaki (18 squares), sea green (23 squares). An even split would hand every color 12 squares, so the amber region is running 11 squares under that average while the sea green region is the roomiest area on the grid with 23 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 amber, with 1 square, is under far more pressure than sea green with 23. A 22-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Amber is the densest region on this grid at 100% of a 1×1 frame; indigo 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
Here is the full footprint of every color on puzzle #186, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Amber: 1 square across rows 2–2, columns 12–12. It never leaves row 2, so its queen is locked to that row.
Orange covers 5 squares, spanning 4 rows and 2 columns (rows 7–10, columns 1–2).
Teal occupies 5 squares within rows 9–12, columns 1–2.
Mint: 7 squares across rows 2–5, columns 10–12.
Indigo covers 10 squares, spanning 4 rows and 6 columns (rows 1–4, columns 7–12).
Slate occupies 10 squares within rows 3–7, columns 1–3.
Pale yellow: 15 squares across rows 3–8, columns 3–6.
Red covers 16 squares, spanning 4 rows and 7 columns (rows 3–6, columns 6–12).
Blue occupies 17 squares within rows 1–3, columns 1–8.
Rose: 17 squares across rows 10–12, columns 6–12.
Khaki covers 18 squares, spanning 4 rows and 6 columns (rows 6–9, columns 7–12).
Sea green occupies 23 squares within rows 8–12, columns 2–8.
Where to start on this board
Amber is the natural opening on #186: it is the smallest region on the grid at 1 square, boxed into rows 2–2 and columns 12–12. 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 amber never leaves row 2, its queen must be in that row — and that instantly settles row 2 for everyone else. Sweep along it and X out every square that is not amber. 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.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: orange and teal span only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #186
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 2 colors. Columns 4 and 5 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 3, meeting 6 colors, is the opposite case — the busiest line on #186 and the last one you should expect to crack.
Working through the middle game
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
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.
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 #186 is rated hard
The rating reflects how much the board concedes up front. Puzzle #186 has two regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 22 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.
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.
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 #186, start hunting for it near row 1.
Here is a shape check that gives nothing away: when #186 is solved, 4 queens sit on the outer edge of the grid and 8 sit inside it, and no queen lands 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 #186?
Puzzle #186 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #186 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #186, 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 #186?
Start with the amber region — the most constrained color on this board, with 1 square spanning rows 2–2 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.