12x12 Queens Puzzle #134
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 30-step reasoning is written out below.
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Step-by-step solution for puzzle #134
What follows is a complete solve of puzzle #134, worked the way a person works it rather than the way a computer does. It runs to 30 deductions — 12 queen placements and 18 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 shadow, pairing argument on columns, proof by contradiction, pairing argument on rows, last square in a row, last square in a column.
Show the full walkthrough (20 steps) — contains the solution
Step 1. The board opens by cascading. Each of these placements is forced the moment the one before it is marked, so the whole run is really a single sweep repeated: indigo has one square left, so its queen takes row 1, column 4; amber has one square left, so its queen takes row 2, column 11; mint has one square left, so its queen takes row 4, column 12; blue has one square left, so its queen takes row 3, column 3; 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.
Step 2. The slate region has 4 squares left, spread across rows 5–7, columns 1 and 2. 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 3. Wherever red 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 4. Here is the pairing argument. The slate and teal regions have no open squares outside columns 1 and 2 — 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 5. Test row 5, column 1 for slate: place a queen there, run the forced consequences through, and row 11 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 5 — 5 of 12 queens placed. Step 6. Now a short proof by contradiction. Suppose the red queen went to row 5, column 5. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 7. Red is down to 3 candidates in rows 5–7, columns 5 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 8. Now a short proof by contradiction. Suppose the red queen went to row 7, column 5. Follow the forced moves that would create and the pale yellow region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 9. Test row 5, column 8 for pale yellow: place a queen there, run the forced consequences through, and row 11 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 10. Now a short proof by contradiction. Suppose the pale yellow queen went to row 5, column 9. Follow the forced moves that would create and row 11 runs out of open squares entirely — impossible. So that square is ruled out for good.
The board after step 10 — 5 of 12 queens placed. Step 11. Test row 5, column 10 for pale yellow: place a queen there, run the forced consequences through, and column 8 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 12. Wherever pale yellow 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 13. Orange is down to 4 candidates in rows 6, 7, 9, and 11, columns 9 and 10, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about orange itself.
Step 14. Now a short proof by contradiction. Suppose the pale yellow queen went to row 6, column 8. Follow the forced moves that would create and row 11 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 15. Test row 7, column 1 for slate: place a queen there, run the forced consequences through, and the rose region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 15 — 5 of 12 queens placed. Step 16. 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: 2 squares come off in one stroke.
Step 17. The orange region has 3 squares left, spread across rows 7, 9, and 11, columns 9 and 10. 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 18. Now a short proof by contradiction. Suppose the pale yellow queen went to row 8, column 8. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 19. Pale yellow is down to 2 candidates in rows 7 and 8, columns 8 and 10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 20. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: row 7 is down to one square, putting a queen on row 7, column 8; column 10 is down to one square, putting a queen on row 9, column 10; column 9 is down to one square, putting a queen on row 12, column 9; row 11 is down to one square, putting a queen on row 11, column 1; slate has one square left, so its queen takes row 5, column 2; red has one square left, so its queen takes row 6, column 5; khaki has one square left, so its queen takes row 8, column 6. 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 8, column 6.
The board after step 20 — 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 #134
Board #134 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, #134 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 #134 are: indigo (1 square), sea green (1 square), amber (5 squares), mint (8 squares), teal (9 squares), slate (10 squares), red (12 squares), rose (16 squares), blue (18 squares), orange (18 squares), pale yellow (22 squares), khaki (24 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the khaki region is the roomiest area on the grid with 24 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 indigo, with 1 square, is under far more pressure than khaki with 24. A 23-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; khaki is the loosest, occupying only 43% of its 7×8 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Below is every color on #134 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Indigo occupies 1 square within rows 1–1, columns 4–4. It never leaves row 1, so its queen is locked to that row.
Sea green: 1 square across rows 10–10, columns 7–7. It never leaves row 10, so its queen is locked to that row.
Amber covers 5 squares, spanning 2 rows and 4 columns (rows 1–2, columns 9–12).
Mint occupies 8 squares within rows 2–5, columns 10–12.
Teal: 9 squares across rows 8–12, columns 1–3.
Slate covers 10 squares, spanning 6 rows and 3 columns (rows 2–7, columns 1–3).
Red occupies 12 squares within rows 3–7, columns 4–6.
Rose: 16 squares across rows 10–12, columns 5–12.
Blue covers 18 squares, spanning 4 rows and 9 columns (rows 1–4, columns 1–9).
Orange occupies 18 squares within rows 5–11, columns 9–12.
Pale yellow: 22 squares across rows 3–8, columns 6–10.
Khaki covers 24 squares, spanning 7 rows and 8 columns (rows 6–12, columns 2–9).
Where to start on this board
Indigo is the natural opening on #134: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 4–4. 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 indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Sea green and amber 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 #134
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 9 and 12 are the narrowest, touched by only 3 colors. Column 1 is the tightest column, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 11 rows. Row 10 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
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.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #134 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers four single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 23. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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 #134, start hunting for it near row 1.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 8 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #134?
Puzzle #134 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #134 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #134, 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 #134?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.