12x12 Queens Puzzle #138
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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #138
What follows is a complete solve of puzzle #138, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 12 queen placements and 7 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: last square in a region, region shadow, region confined to one line, pairing argument on rows, pairing argument on columns, last square in a row.
Show the full walkthrough (11 steps) — contains the solution
Step 1. Only row 7, column 12 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 12, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 2. The rose region is down to one open square, at row 12, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 12 and column 6, the remaining rose squares, and the ring of neighbours around row 12, column 6.
Step 3. Only row 10, column 7 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 7, in sea green and in the ring around that square is now dead.
Step 4. Wherever blue 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.
The board after step 4 — 3 of 12 queens placed. Step 5. Slate cannot leave row 5 — every one of its open squares (2 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 10 squares in all.
Step 6. The orange region has 3 squares left, spread across rows 8, 9, and 11, columns 4, 5, and 8. 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. Teal cannot leave row 11 — every one of its open squares (3 open squares) sits there. That row belongs to teal, so every square on it owned by another colour is dead: 4 squares in all.
The board after step 7 — 3 of 12 queens placed. Step 8. Orange is down to 2 candidates in rows 8 and 9, columns 5 and 8, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about orange itself.
Step 9. Indigo, amber, blue, and mint are now boxed into rows 1–4 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 8 squares at once.
Step 10. Here is the pairing argument. The indigo, mint, slate, and khaki regions have no open squares outside columns 8–11 — 4 colours confined to 4 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 1 square come off in one stroke.
Step 11. 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: orange has one square left, so its queen takes row 9, column 5; row 4 is down to one square, putting a queen on row 4, column 9; slate has one square left, so its queen takes row 5, column 11; row 3 is down to one square, putting a queen on row 3, column 1; row 2 is down to one square, putting a queen on row 2, column 3; teal has one square left, so its queen takes row 11, column 2; red has one square left, so its queen takes row 6, column 4; khaki has one square left, so its queen takes row 8, column 10; indigo has one square left, so its queen takes row 1, column 8. 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 1, column 8.
The board after step 11 — 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 #138
Board #138 of the 12x12 bank carries a medium 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, #138 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 #138 are: pale yellow (1 square), rose (1 square), sea green (2 squares), blue (3 squares), slate (5 squares), indigo (10 squares), amber (10 squares), teal (10 squares), mint (14 squares), orange (18 squares), khaki (32 squares), red (38 squares). An even split would hand every color 12 squares, so the pale yellow region is running 11 squares under that average while the red 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 pale yellow, with 1 square, is under far more pressure than red with 38. A 37-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Pale yellow is the densest region on this grid at 100% of a 1×1 frame; amber is the loosest, occupying only 42% of its 4×6 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Here is the full footprint of every color on puzzle #138, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Pale yellow: 1 square across rows 7–7, columns 12–12. It never leaves row 7, so its queen is locked to that row.
Rose covers 1 square, spanning 1 row and 1 column (rows 12–12, columns 6–6). It never leaves row 12, so its queen is locked to that row.
Sea green occupies 2 squares within rows 10–10, columns 6–7. It never leaves row 10, so its queen is locked to that row.
Blue: 3 squares across rows 2–3, columns 1–2.
Slate covers 5 squares, spanning 3 rows and 3 columns (rows 4–6, columns 10–12).
Indigo occupies 10 squares within rows 1–3, columns 6–11.
Amber: 10 squares across rows 1–4, columns 1–6.
Teal covers 10 squares, spanning 3 rows and 4 columns (rows 10–12, columns 1–4).
Mint occupies 14 squares within rows 1–5, columns 8–12.
Orange: 18 squares across rows 7–12, columns 4–8.
Khaki covers 32 squares, spanning 9 rows and 6 columns (rows 4–12, columns 7–12).
Red occupies 38 squares within rows 2–9, columns 1–7.
Where to start on this board
Pale yellow is the natural opening on #138: it is the smallest region on the grid at 1 square, boxed into rows 7–7 and columns 12–12. 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 pale yellow never leaves row 7, its queen must be in that row — and that instantly settles row 7 for everyone else. Sweep along it and X out every square that is not pale yellow. 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, rose and blue 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 #138
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 6, 8, 9 and 11 are the narrowest, touched by only 3 colors. Columns 3, 5 and 9 are the tightest columns, with 3 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 11 rows. By contrast row 4 meets 5 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
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.
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 #138 is rated medium
The rating reflects how much the board concedes up front. Puzzle #138 has five regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 37 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
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 #138, start hunting for it near row 1.
As a sanity check rather than a spoiler: the finished solution to #138 puts 4 queens on the outer border and 8 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 #138?
Puzzle #138 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #138 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #138, 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 #138?
Start with the pale yellow region — the most constrained color on this board, with 1 square spanning rows 7–7 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.