12x12 Queens Puzzle #173
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 #173
What follows is a complete solve of puzzle #173, 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 columns.
Techniques this board needs: region shadow, pairing argument on rows, pairing argument on columns, region confined to one line, last square in a column, line confined to one region, last square in a region.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Wherever red ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 2. Rose is down to 3 candidates in rows 11 and 12, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about rose itself.
Step 3. Indigo and amber are now boxed into rows 1 and 2 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 4. Wherever blue ends up among its 5 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 — 0 of 12 queens placed. Step 5. Red and rose 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 11 squares at once.
Step 6. All of the slate region's remaining squares (2 open squares) lie in column 3. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 13 squares come off.
Step 7. Scan column 2 and only row 12, column 2 remains — a rose square. Every column takes a queen, so it takes this one. Cross out the whole of row 12 and column 2, the remaining rose squares, and the ring of neighbours around row 12, column 2.
Step 8. All of the indigo region's remaining squares (3 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.
The board after step 8 — 1 of 12 queens placed. Step 9. Blue cannot leave row 3 — every one of its open squares (2 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 8 squares in all.
Step 10. Every open square left in row 4 (7 open squares) belongs to the mint region. Since that row must contain a queen, mint is the colour that supplies it — so mint squares anywhere else on the board are out, 23 squares in all.
Step 11. Indigo, blue, pale yellow, and khaki are now boxed into columns 4–7 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 6 squares at once.
Step 12. From here the board cascades. Keep alternating the region sweep and the line sweep and 11 queens fall out in order without any new idea being needed: sea green has one square left, so its queen takes row 10, column 8; teal has one square left, so its queen takes row 11, column 12; orange has one square left, so its queen takes row 9, column 11; amber has one square left, so its queen takes row 2, column 10; mint has one square left, so its queen takes row 4, column 9; khaki has one square left, so its queen takes row 8, column 5; blue has one square left, so its queen takes row 3, column 4; indigo has one square left, so its queen takes row 1, column 6; pale yellow has one square left, so its queen takes row 7, column 7; red has one square left, so its queen takes row 6, column 1; slate has one square left, so its queen takes row 5, column 3. 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 5, column 3.
The board after step 12 — 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 #173
Puzzle #173 is one of the medium-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, #173 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 #173 are: rose (3 squares), red (4 squares), sea green (5 squares), indigo (7 squares), orange (8 squares), blue (9 squares), slate (9 squares), amber (10 squares), teal (11 squares), pale yellow (16 squares), khaki (24 squares), mint (38 squares). An even split would hand every color 12 squares, so the rose region is running 9 squares under that average while the mint region is the roomiest area on the grid with 38 squares.
Every color takes one queen and one only, whatever its size — which is why the 35 square difference between rose and mint 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. Amber is the tidiest shape on #173, packing 83% of its 2×6 bounding box, whereas orange straggles across its 6×3 box at just 44%. 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.
Rose occupies 3 squares within rows 11–12, columns 1–2.
Red: 4 squares across rows 6–8, columns 1–2.
Sea green covers 5 squares, spanning 2 rows and 3 columns (rows 9–10, columns 6–8).
Indigo occupies 7 squares within rows 1–2, columns 2–6.
Orange: 8 squares across rows 6–11, columns 10–12.
Blue covers 9 squares, spanning 3 rows and 6 columns (rows 2–4, columns 2–7).
Slate occupies 9 squares within rows 1–6, columns 1–3.
Amber: 10 squares across rows 1–2, columns 7–12.
Teal covers 11 squares, spanning 4 rows and 5 columns (rows 9–12, columns 8–12).
Pale yellow occupies 16 squares within rows 4–8, columns 2–7.
Khaki: 24 squares across rows 8–12, columns 1–8.
Mint covers 38 squares, spanning 8 rows and 7 columns (rows 2–9, columns 6–12).
Where to start on this board
Open on the tightest color. Here that is rose, confined to 3 squares between rows 11 and 12 and columns 1 and 2. Its queen has nowhere else to go, which means those 2 rows and 2 columns carry more constraint than anywhere else on #173.
Rose does not collapse onto a single line, so it will not hand you a queen on move one. What it does give you is a short list of 3 candidates. Take each one in turn, imagine the queen there, cross out its row, its column and the eight squares around it, and ask whether any other region is left with nowhere to go. One or two candidates usually fail that test immediately.
Work outward to the other flat regions next. Indigo, amber and sea green 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 #173
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 3, 5 and 12 are the narrowest, touched by only 3 colors. Column 9 is the tightest column, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 9, meeting 5 colors, is the opposite case — the busiest line on #173 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.
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 #173 is rated medium
Ratings track how generous the opening is. On #173 you get no region pinned to a single row or column, a smallest color of 3 squares, a tightest row crossed by 3 colors, and a 35-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
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.
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 #173 is row 1.
Here is a shape check that gives nothing away: when #173 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 #173?
Puzzle #173 is rated medium. Its smallest color region holds 3 squares, its tightest row is touched by 3 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 12x12 puzzle #173 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #173, 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 #173?
Start with the rose region — the most constrained color on this board, with 3 squares spanning rows 11–12 and columns 1–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.