12x12 Queens Puzzle #175
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 22-step reasoning is written out below.
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Step-by-step solution for puzzle #175
What follows is a complete solve of puzzle #175, worked the way a person works it rather than the way a computer does. It runs to 22 deductions — 12 queen placements and 10 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 rows.
Techniques this board needs: region confined to one line, region shadow, pairing argument on rows, last square in a region.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Sea green cannot leave column 1 — every one of its open squares (3 open squares) sits there. That column belongs to sea green, so every square in it owned by another colour is dead: 10 squares in all.
Step 2. Indigo is down to 5 candidates in rows 1–3, columns 2–4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. Red cannot leave column 2 — every one of its open squares (3 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 9 squares in all.
Step 4. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 9 squares come off.
The board after step 4 — 0 of 12 queens placed. Step 5. Amber is down to 4 candidates in rows 2–4, columns 8 and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 6. All of the teal region's remaining squares (2 open squares) lie in column 12. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 11 squares come off.
Step 7. Wherever orange ends up among its 6 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 8. Here is the pairing argument. The sea green, teal, and rose regions have no open squares outside rows 10–12 — 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. Only row 9, column 11 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 11, the rest of the orange region and every square adjacent to it — diagonals included.
The board after step 9 — 1 of 12 queens placed. Step 10. The khaki region is down to one open square, at row 8, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 4, the remaining khaki squares, and the ring of neighbours around row 8, column 4.
Step 11. Only row 1, column 3 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 3, in indigo and in the ring around that square is now dead.
Step 12. All of the blue region's remaining squares (2 open squares) lie in column 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 6 squares come off.
Step 13. Only row 7, column 9 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 9, the rest of the pale yellow region and every square adjacent to it — diagonals included.
The board after step 13 — 4 of 12 queens placed. Step 14. The amber region is down to one open square, at row 2, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 8, the remaining amber squares, and the ring of neighbours around row 2, column 8.
Step 15. Only row 3, column 10 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 10, in blue and in the ring around that square is now dead.
Step 16. Wherever mint ends up among its 7 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 17. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: slate has one square left, so its queen takes row 5, column 7; red has one square left, so its queen takes row 6, column 2; mint has one square left, so its queen takes row 4, column 5; rose has one square left, so its queen takes row 12, column 6; teal has one square left, so its queen takes row 11, column 12; sea green has one square left, so its queen takes row 10, column 1. 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 1.
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 #175
12x12 puzzle #175 is rated medium. Twelve colored regions cover its 144 squares, and your job is to seat twelve queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
There is exactly one way to finish #175, 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 #175 are: teal (2 squares), sea green (3 squares), indigo (8 squares), amber (8 squares), red (8 squares), blue (11 squares), orange (11 squares), slate (12 squares), pale yellow (15 squares), khaki (16 squares), rose (16 squares), mint (34 squares). An even split would hand every color 12 squares, so the teal region is running 10 squares under that average while the mint region is the roomiest area on the grid with 34 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Teal has only 2 places to put its queen; mint has 34. The spread between them on this board is 32 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
A region's outline can constrain it more than its size does. Here sea green is the most solid, filling 100% of its 3×1 box, and slate the most diffuse at 48% of its 5×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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 medium board.
Teal covers 2 squares, spanning 2 rows and 1 column (rows 11–12, columns 12–12). It sits wholly inside column 12, so its queen must be in that column.
Sea green occupies 3 squares within rows 10–12, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Indigo: 8 squares across rows 1–3, columns 1–4.
Amber covers 8 squares, spanning 4 rows and 4 columns (rows 1–4, columns 7–10).
Red occupies 8 squares within rows 4–8, columns 1–2.
Blue: 11 squares across rows 1–5, columns 10–12.
Orange covers 11 squares, spanning 5 rows and 4 columns (rows 8–12, columns 9–12).
Slate occupies 12 squares within rows 4–8, columns 6–10.
Pale yellow: 15 squares across rows 5–9, columns 8–12.
Khaki covers 16 squares, spanning 4 rows and 7 columns (rows 8–11, columns 1–7).
Rose occupies 16 squares within rows 10–12, columns 2–9.
Mint: 34 squares across rows 1–8, columns 2–8.
Where to start on this board
Begin where the choices are fewest. On this board that means teal, which owns just 2 squares inside a 2×1 window running from row 11 to row 12 and column 12 to column 12. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because teal stays inside column 12, its queen belongs to that column and no other region may use it. Cross out the rest of column 12 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: red spans 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 #175
A quick way to find the next move is to count how many different colors touch each line. On this board, row 9 is the narrowest, touched by only 3 colors. Columns 5 and 11 are the tightest columns, with 3 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 8 is the crowded extreme at 6 colors — worth revisiting later, not now.
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.
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.
Resist the plausible-looking queen. On #175 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #175 is rated medium
The medium label comes out of four measurable things about #175: two of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 32. Boards that score tight on those measures give up their first queen fast.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
Stuck on #175 despite the medium 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.
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 #175 the place to look for one is around row 9.
A non-spoiling distribution check: solved, #175 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 #175?
Puzzle #175 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #175 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #175, 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 #175?
Start with the teal region — the most constrained color on this board, with 2 squares spanning rows 11–12 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.