12x12 Queens Puzzle #150
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 20-step reasoning is written out below.
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Step-by-step solution for puzzle #150
What follows is a complete solve of puzzle #150, worked the way a person works it rather than the way a computer does. It runs to 20 deductions — 12 queen placements and 8 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: region confined to one line, last square in a region, proof by contradiction.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Amber cannot leave column 9 — 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: 14 squares in all.
Step 2. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: mint has one square left, so its queen takes row 4, column 10; amber has one square left, so its queen takes row 2, column 9; indigo has one square left, so its queen takes row 1, column 7; blue has one square left, so its queen takes row 3, column 12. 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 3, column 12.
Step 3. All of the slate region's remaining squares (2 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 7 squares come off.
Step 4. 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: 4 squares in all.
Step 5. The pale yellow region is down to one open square, at row 7, column 1. Every colour holds exactly one queen, so that square is forced. Row 7 and column 1 are now spoken for, along with the squares touching that queen.
The board after step 5 — 5 of 12 queens placed. Step 6. Only row 5, column 2 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 2, the rest of the slate region and every square adjacent to it — diagonals included.
Step 7. The khaki region is down to one open square, at row 8, column 11. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 11, the remaining khaki squares, and the ring of neighbours around row 8, column 11.
Step 8. Rose cannot leave row 12 — every one of its open squares (2 open squares) sits there. That row belongs to rose, so every square on it owned by another colour is dead: 5 squares in all.
Step 9. All of the sea green region's remaining squares (2 open squares) lie in row 10. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.
The board after step 9 — 7 of 12 queens placed. Step 10. Teal cannot leave row 11 — every one of its open squares (2 open squares) sits there. That row belongs to teal, so every square on it owned by another colour is dead: 1 square in all.
Step 11. Now a short proof by contradiction. Suppose the red queen went to row 6, column 8. Follow the forced moves that would create and column 4 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 12. Only row 6, column 6 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 6, in red and in the ring around that square is now dead.
Step 13. Now a short proof by contradiction. Suppose the orange queen went to row 9, column 8. Follow the forced moves that would create and column 4 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: orange has one square left, so its queen takes row 9, column 5; sea green has one square left, so its queen takes row 10, column 3; teal has one square left, so its queen takes row 11, column 8; rose has one square left, so its queen takes row 12, 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 12, column 4.
The board after step 14 — 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 #150
Board #150 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.
Like every board on this site, #150 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.
How the colors divide this board
From smallest to largest, the regions on puzzle #150 are: amber (2 squares), mint (2 squares), rose (2 squares), pale yellow (5 squares), indigo (9 squares), blue (16 squares), slate (17 squares), khaki (17 squares), teal (17 squares), red (18 squares), sea green (18 squares), orange (21 squares). An even split would hand every color 12 squares, so the amber region is running 10 squares under that average while the orange region is the roomiest area on the grid with 21 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 orange with 21. A 19-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Amber is the densest region on this grid at 100% of a 2×1 frame; indigo is the loosest, occupying only 45% of its 4×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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 hard board.
Amber: 2 squares across rows 2–3, columns 9–9. It sits wholly inside column 9, so its queen must be in that column.
Mint covers 2 squares, spanning 1 row and 2 columns (rows 4–4, columns 9–10). It never leaves row 4, so its queen is locked to that row.
Rose occupies 2 squares within rows 12–12, columns 3–4. It never leaves row 12, so its queen is locked to that row.
Pale yellow: 5 squares across rows 6–7, columns 1–4.
Indigo covers 9 squares, spanning 4 rows and 5 columns (rows 1–4, columns 4–8).
Blue occupies 16 squares within rows 1–6, columns 9–12.
Slate: 17 squares across rows 1–5, columns 1–6.
Khaki covers 17 squares, spanning 7 rows and 5 columns (rows 5–11, columns 8–12).
Teal occupies 17 squares within rows 9–12, columns 5–12.
Red: 18 squares across rows 3–6, columns 3–9.
Sea green covers 18 squares, spanning 6 rows and 4 columns (rows 7–12, columns 1–4).
Orange occupies 21 squares within rows 6–11, columns 3–9.
Where to start on this board
Amber is the natural opening on #150: it is the smallest region on the grid at 2 squares, boxed into rows 2–3 and columns 9–9. Since one of those squares must hold a queen, that block of 2 rows and 1 column is the most contested part of the board.
Because amber stays inside column 9, its queen belongs to that column and no other region may use it. Cross out the rest of column 9 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. Mint, rose and pale yellow 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 #150
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 8 and 12 are the narrowest, touched by only 3 colors. Columns 1, 2, 11 and 12 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 6 is the crowded extreme at 5 colors — worth revisiting later, not now.
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.
With twelve queens to seat and twelve regions competing over 144 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 12x12 boards quickly and players who stall halfway.
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 #150 is rated hard
The hard label comes out of four measurable things about #150: three 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 19. 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.
If #150 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.
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 #150, start hunting for it near row 1.
A non-spoiling distribution check: solved, #150 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 #150?
Puzzle #150 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #150 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #150, 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 #150?
Start with the amber region — the most constrained color on this board, with 2 squares spanning rows 2–3 and columns 9–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.