12x12 Queens Puzzle #29
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 #29
What follows is a complete solve of puzzle #29, 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 pairing argument on columns.
Techniques this board needs: region confined to one line, region shadow, last square in a region, pairing argument on rows, pairing argument on columns, last square in a row, last square in a column.
Show the full walkthrough (15 steps) — contains the solution
Step 1. Amber cannot leave column 12 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 12 squares in all.
Step 2. Slate is down to 5 candidates in rows 4–6, columns 8 and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.
Step 3. Red cannot leave row 6 — every one of its open squares (2 open squares) sits there. That row belongs to red, so every square on it owned by another colour is dead: 13 squares in all.
Step 4. Wherever slate 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.
Step 5. Khaki is down to 3 candidates in rows 7 and 8, columns 1 and 2, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about khaki itself.
The board after step 5 — 0 of 12 queens placed. Step 6. The orange region has 4 squares left, spread across rows 8 and 9, columns 6–8. 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 7. Teal cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to teal, so every square in it owned by another colour is dead: 10 squares in all.
Step 8. The khaki region is down to one open square, at row 8, column 2. Every colour holds exactly one queen, so that square is forced. Row 8 and column 2 are now spoken for, along with the squares touching that queen.
Step 9. 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: 7 squares in all.
Step 10. Here is the pairing argument. The teal and rose regions have no open squares outside rows 11 and 12 — 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: 12 squares come off in one stroke.
The board after step 10 — 1 of 12 queens placed. Step 11. Indigo, slate, red, and sea green are now boxed into columns 8–11 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. The orange region is down to one open square, at row 9, column 7. Every colour holds exactly one queen, so that square is forced. Row 9 and column 7 are now spoken for, along with the squares touching that queen.
Step 13. Look along row 11: one square survives, row 11, column 1, which belongs to teal. A row cannot be left empty, so the queen goes there. Everything else in row 11, in column 1, in teal and in the ring around that square is now dead.
Step 14. Here is the pairing argument. The indigo, amber, blue, and mint regions have no open squares outside rows 1–4 — 4 colours confined to 4 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 3 squares come off in one stroke.
Step 15. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: slate has one square left, so its queen takes row 5, column 9; red has one square left, so its queen takes row 6, column 11; sea green has one square left, so its queen takes row 10, column 10; indigo has one square left, so its queen takes row 1, column 8; amber has one square left, so its queen takes row 2, column 12; column 3 is down to one square, putting a queen on row 3, column 3; mint has one square left, so its queen takes row 4, column 6; rose has one square left, so its queen takes row 12, column 5; pale yellow has one square left, so its queen takes row 7, column 4. 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 7, column 4.
The board after step 15 — 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 #29
Puzzle #29 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, #29 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 #29 are: amber (2 squares), teal (2 squares), red (3 squares), orange (4 squares), slate (5 squares), khaki (5 squares), rose (6 squares), indigo (17 squares), mint (17 squares), blue (19 squares), sea green (26 squares), pale yellow (38 squares). An even split would hand every color 12 squares, so the amber region is running 10 squares under that average while the pale yellow region is the roomiest area on the grid with 38 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 2 squares, is under far more pressure than pale yellow with 38. A 36-square gap like this one rewards players who start where the choices are fewest.
Square count is only half the story — how a color is arranged counts too. Amber is the tidiest shape on #29, packing 100% of its 2×1 bounding box, whereas rose straggles across its 2×5 box at just 60%. 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 medium board.
Amber occupies 2 squares within rows 1–2, columns 12–12. It sits wholly inside column 12, so its queen must be in that column.
Teal: 2 squares across rows 11–12, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Red covers 3 squares, spanning 1 row and 3 columns (rows 6–6, columns 10–12). It never leaves row 6, so its queen is locked to that row.
Orange occupies 4 squares within rows 8–9, columns 6–8.
Slate: 5 squares across rows 4–6, columns 8–9.
Khaki covers 5 squares, spanning 3 rows and 2 columns (rows 6–8, columns 1–2).
Rose occupies 6 squares within rows 11–12, columns 5–9.
Indigo: 17 squares across rows 1–5, columns 8–12.
Mint covers 17 squares, spanning 7 rows and 4 columns (rows 1–7, columns 5–8).
Blue occupies 19 squares within rows 1–5, columns 1–5.
Sea green: 26 squares across rows 7–12, columns 8–12.
Pale yellow covers 38 squares, spanning 9 rows and 7 columns (rows 4–12, columns 1–7).
Where to start on this board
Open on the tightest color. Here that is amber, confined to 2 squares between rows 1 and 2 and columns 12 and 12. Its queen has nowhere else to go, which means those 2 rows and 1 column carry more constraint than anywhere else on #29.
Because amber 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.
Work outward to the other flat regions next. Red, orange, teal 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 #29
A quick way to find the next move is to count how many different colors touch each line. On this board, row 10 is the narrowest, touched by only 2 colors. Columns 3 and 4 are the tightest columns, with 2 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 6, meeting 5 colors, is the opposite case — the busiest line on #29 and the last one you should expect to crack.
Working through the middle game
Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.
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.
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 #29 is rated medium
The medium label comes out of four measurable things about #29: three of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 2 colors, and its largest and smallest regions differ by 36. Boards that score tight on those measures give up their first queen fast.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
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 #29, start hunting for it near row 10.
Here is a shape check that gives nothing away: when #29 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 #29?
Puzzle #29 is rated medium. Its smallest color region holds 2 squares, 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 #29 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #29, 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 #29?
Start with the amber region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.