12x12 Queens Puzzle #37
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 22-step reasoning is written out below.
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Step-by-step solution for puzzle #37
What follows is a complete solve of puzzle #37, 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 confined to one line, region shadow, pairing argument on rows, last square in a region, pairing argument on columns.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 12 squares in all.
Step 2. Amber is down to 4 candidates in rows 2–4, columns 1–3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 3. Slate cannot leave column 12 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 11 squares in all.
Step 4. Wherever teal 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. Rose is down to 5 candidates in rows 10–12, columns 8–10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about rose itself.
The board after step 5 — 0 of 12 queens placed. Step 6. Here is the pairing argument. The amber, blue, and mint regions have no open squares outside rows 2–4 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 10 squares come off in one stroke.
Step 7. Only row 5, column 12 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 12, in slate and in the ring around that square is now dead.
Step 8. The red region is down to one open square, at row 6, column 8. Every colour holds exactly one queen, so that square is forced. Row 6 and column 8 are now spoken for, along with the squares touching that queen.
Step 9. The khaki region has 6 squares left, spread across rows 7, 8, 10, 11, and 12, columns 10 and 11. 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.
The board after step 9 — 2 of 12 queens placed. Step 10. Here is the pairing argument. The indigo and teal regions have no open squares outside columns 3 and 4 — 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: 11 squares come off in one stroke.
Step 11. Amber cannot leave row 2 — every one of its open squares (2 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 6 squares in all.
Step 12. The blue region is down to one open square, at row 3, column 11. Every colour holds exactly one queen, so that square is forced. Row 3 and column 11 are now spoken for, along with the squares touching that queen.
Step 13. Khaki cannot leave column 10 — every one of its open squares (2 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 5 squares in all.
Step 14. 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: rose has one square left, so its queen takes row 12, column 9; teal has one square left, so its queen takes row 11, column 4; indigo has one square left, so its queen takes row 1, column 3; amber has one square left, so its queen takes row 2, column 1; sea green has one square left, so its queen takes row 10, column 2; pale yellow has one square left, so its queen takes row 7, column 5; mint has one square left, so its queen takes row 4, column 6; khaki has one square left, so its queen takes row 8, column 10; orange has one square left, so its queen takes row 9, 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 9, column 7.
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 #37
Puzzle #37 is one of the hard-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, #37 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 #37 are: indigo (2 squares), slate (2 squares), teal (3 squares), rose (5 squares), amber (8 squares), mint (12 squares), red (12 squares), sea green (13 squares), blue (15 squares), khaki (17 squares), pale yellow (26 squares), orange (29 squares). An even split would hand every color 12 squares, so the indigo region is running 10 squares under that average while the orange region is the roomiest area on the grid with 29 squares.
Every color takes one queen and one only, whatever its size — which is why the 27 square difference between indigo and orange 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. Indigo is the tidiest shape on #37, packing 100% of its 1×2 bounding box, whereas pale yellow straggles across its 7×8 box at just 46%. 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 hard board.
Indigo covers 2 squares, spanning 1 row and 2 columns (rows 1–1, columns 3–4). It never leaves row 1, so its queen is locked to that row.
Slate occupies 2 squares within rows 4–5, columns 12–12. It sits wholly inside column 12, so its queen must be in that column.
Teal: 3 squares across rows 11–12, columns 3–4.
Rose covers 5 squares, spanning 3 rows and 3 columns (rows 10–12, columns 8–10).
Amber occupies 8 squares within rows 1–4, columns 1–4.
Mint: 12 squares across rows 1–4, columns 4–7.
Red covers 12 squares, spanning 4 rows and 6 columns (rows 3–6, columns 6–11).
Sea green occupies 13 squares within rows 7–12, columns 1–4.
Blue: 15 squares across rows 1–4, columns 8–12.
Khaki covers 17 squares, spanning 7 rows and 4 columns (rows 6–12, columns 9–12).
Pale yellow occupies 26 squares within rows 3–9, columns 1–8.
Orange: 29 squares across rows 7–12, columns 4–11.
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 2 squares between rows 1 and 1 and columns 3 and 4. Its queen has nowhere else to go, which means those 1 row and 2 columns carry more constraint than anywhere else on #37.
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. Slate and teal 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 #37
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 5 and 6 are the narrowest, touched by only 3 colors. Columns 1, 2, 5 and 12 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 6 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.
The whole middle game reduces to two sweeps. Sweep the twelve colors looking for one with a single square left; sweep the twelve rows and twelve columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 12x12 solve from a stalled one.
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 #37 is rated hard
The hard label comes out of four measurable things about #37: two 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 27. Boards that score tight on those measures give up their first queen fast.
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 #37 is row 2.
As a sanity check rather than a spoiler: the finished solution to #37 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 #37?
Puzzle #37 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #37 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #37, 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 #37?
Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–1 and columns 3–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.