12x12 Queens Puzzle #161
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 24-step reasoning is written out below.
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Step-by-step solution for puzzle #161
What follows is a complete solve of puzzle #161, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 12 queen placements and 12 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, pairing argument on columns, proof by contradiction, last square in a row.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Only row 12, column 3 is still open in rose, and since each colour must take a queen, that settles it. That clears row 12, column 3, the rest of the rose region and every square adjacent to it — diagonals included.
Step 2. The teal region is down to one open square, at row 11, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 1, the remaining teal squares, and the ring of neighbours around row 11, column 1.
Step 3. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 10 squares in all.
Step 4. All of the blue region's remaining squares (2 open squares) lie in column 8. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 10 squares come off.
The board after step 4 — 2 of 12 queens placed. Step 5. Red is down to 7 candidates in rows 6–9, columns 5–7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 6. The sea green region has 5 squares left, spread across rows 8–10, columns 5–7. 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. Amber and blue are now boxed into rows 2 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 7 squares at once.
Step 8. Here is the pairing argument. The khaki, orange, and sea green regions have no open squares outside rows 8–10 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 11 squares come off in one stroke.
Step 9. The pale yellow region has 5 squares left, spread across rows 5–7, columns 9–11. 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.
The board after step 9 — 2 of 12 queens placed. Step 10. Here is the pairing argument. The mint and khaki regions have no open squares outside columns 2 and 4 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 3 squares come off in one stroke.
Step 11. Amber, red, and sea green are now boxed into columns 5–7 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 8 squares at once.
Step 12. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 10. Follow the forced moves that would create and the orange region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 13. Only row 1, column 9 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 9, the rest of the indigo region and every square adjacent to it — diagonals included.
The board after step 13 — 3 of 12 queens placed. Step 14. The blue region is down to one open square, at row 3, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 8, the remaining blue squares, and the ring of neighbours around row 3, column 8.
Step 15. The pale yellow region has 4 squares left, spread across rows 5–7, columns 10 and 11. 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 16. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 6. Follow the forced moves that would create and column 4 runs out of open squares entirely — 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 8 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 5; row 8 is down to one square, putting a queen on row 8, column 2; row 9 is down to one square, putting a queen on row 9, column 10; pale yellow has one square left, so its queen takes row 7, column 11; red has one square left, so its queen takes row 6, column 6; mint has one square left, so its queen takes row 4, column 4; slate has one square left, so its queen takes row 5, column 12; sea green has one square left, so its queen takes row 10, column 7. 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 10, column 7.
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 #161
Puzzle #161 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.
As with every puzzle in this collection, #161 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 #161 are: rose (1 square), indigo (2 squares), blue (4 squares), teal (4 squares), red (7 squares), khaki (10 squares), orange (12 squares), amber (15 squares), pale yellow (15 squares), sea green (20 squares), mint (24 squares), slate (30 squares). An even split would hand every color 12 squares, so the rose region is running 11 squares under that average while the slate region is the roomiest area on the grid with 30 squares.
Every color takes one queen and one only, whatever its size — which is why the 29 square difference between rose 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.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #161, packing 100% of its 1×2 bounding box, whereas slate straggles across its 7×9 box at just 48%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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.
Rose occupies 1 square within rows 12–12, columns 3–3. It never leaves row 12, so its queen is locked to that row.
Indigo: 2 squares across rows 1–1, columns 9–10. It never leaves row 1, so its queen is locked to that row.
Blue covers 4 squares, spanning 3 rows and 2 columns (rows 1–3, columns 8–9).
Teal occupies 4 squares within rows 11–12, columns 1–2.
Red: 7 squares across rows 6–9, columns 5–7.
Khaki covers 10 squares, spanning 5 rows and 4 columns (rows 7–11, columns 1–4).
Orange occupies 12 squares within rows 8–12, columns 9–12.
Amber: 15 squares across rows 1–3, columns 2–7.
Pale yellow covers 15 squares, spanning 6 rows and 5 columns (rows 5–10, columns 8–12).
Sea green occupies 20 squares within rows 8–12, columns 4–9.
Mint: 24 squares across rows 1–9, columns 1–4.
Slate covers 30 squares, spanning 7 rows and 9 columns (rows 1–7, columns 4–12).
Where to start on this board
Open on the tightest color. Here that is rose, confined to 1 square between rows 12 and 12 and columns 3 and 3. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #161.
Because rose never leaves row 12, its queen must be in that row — and that instantly settles row 12 for everyone else. Sweep along it and X out every square that is not rose. 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. Indigo and teal 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 #161
A quick way to find the next move is to count how many different colors touch each line. On this board, row 4 is the narrowest, touched by only 2 colors. Columns 1, 11 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 2 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 #161 and the last one you should expect to crack.
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.
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 #161 is rated easy
The easy label comes out of four measurable things about #161: three of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 29. 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.
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.
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 #161 is row 4.
Here is a shape check that gives nothing away: when #161 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 #161?
Puzzle #161 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #161 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #161, 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 #161?
Start with the rose region — the most constrained color on this board, with 1 square spanning rows 12–12 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.