12x12 Queens Puzzle #13
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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #13
What follows is a complete solve of puzzle #13, 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 rows.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, line confined to one region, last square in a column.
Show the full walkthrough (10 steps) — contains the solution
Step 1. Only row 11, column 12 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 12, the rest of the teal region and every square adjacent to it — diagonals included.
Step 2. All of the rose region's remaining squares (2 open squares) lie in row 12. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.
Step 3. The orange region has 4 squares left, spread across rows 8–10, 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 3 — 1 of 12 queens placed. Step 4. All of the sea green region's remaining squares (2 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 12 squares come off.
Step 5. Indigo and amber are now boxed into rows 1 and 2 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 9 squares at once.
Step 6. All of the blue region's remaining squares (2 open squares) lie in row 3. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 10 squares come off.
Step 7. Only row 4, column 7 is still open in mint, and since each colour must take a queen, that settles it. Everything else in row 4, in column 7, in mint and in the ring around that square is now dead.
The board after step 7 — 2 of 12 queens placed. Step 8. Red is down to 3 candidates in rows 6 and 7, columns 6 and 8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 9. Column 5 has only khaki squares left (4 open squares). The column needs a queen and only khaki can provide one, which pins that colour to this column and kills its squares elsewhere: 16 squares.
Step 10. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: column 6 is down to one square, putting a queen on row 6, column 6; khaki has one square left, so its queen takes row 8, column 5; sea green has one square left, so its queen takes row 10, column 4; orange has one square left, so its queen takes row 9, column 11; blue has one square left, so its queen takes row 3, column 10; slate has one square left, so its queen takes row 5, column 9; pale yellow has one square left, so its queen takes row 7, column 1; rose has one square left, so its queen takes row 12, column 2; 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 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 2, column 8.
The board after step 10 — 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 #13
Puzzle #13 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.
There is exactly one way to finish #13, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these twelve regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #13 are: teal (1 square), rose (5 squares), indigo (6 squares), amber (6 squares), sea green (6 squares), blue (7 squares), red (8 squares), slate (14 squares), mint (15 squares), orange (17 squares), pale yellow (18 squares), khaki (41 squares). An even split would hand every color 12 squares, so the teal region is running 11 squares under that average while the khaki region is the roomiest area on the grid with 41 squares.
Every color takes one queen and one only, whatever its size — which is why the 40 square difference between teal and khaki 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. Teal is the tidiest shape on #13, packing 100% of its 1×1 bounding box, whereas sea green straggles across its 4×3 box at just 50%. 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.
Teal covers 1 square, spanning 1 row and 1 column (rows 11–11, columns 12–12). It never leaves row 11, so its queen is locked to that row.
Rose occupies 5 squares within rows 11–12, columns 1–3.
Indigo: 6 squares across rows 1–2, columns 1–4.
Amber covers 6 squares, spanning 2 rows and 4 columns (rows 1–2, columns 8–11).
Sea green occupies 6 squares within rows 9–12, columns 3–5.
Blue: 7 squares across rows 1–3, columns 10–12.
Red covers 8 squares, spanning 3 rows and 3 columns (rows 5–7, columns 6–8).
Slate occupies 14 squares within rows 4–7, columns 8–12.
Mint: 15 squares across rows 1–4, columns 3–9.
Orange covers 17 squares, spanning 6 rows and 5 columns (rows 7–12, columns 8–12).
Pale yellow occupies 18 squares within rows 3–10, columns 1–3.
Khaki: 41 squares across rows 4–12, columns 2–10.
Where to start on this board
Open on the tightest color. Here that is teal, confined to 1 square between rows 11 and 11 and columns 12 and 12. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #13.
Because teal never leaves row 11, its queen must be in that row — and that instantly settles row 11 for everyone else. Sweep along it and X out every square that is not teal. 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. Indigo, amber 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 #13
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 8 are the narrowest, touched by only 3 colors. Columns 1, 5, 6 and 7 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 11 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
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
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 #13 is rated hard
The hard label comes out of four measurable things about #13: two of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 40. 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.
Stuck on #13 despite the hard rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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 #13 is row 3.
As a sanity check rather than a spoiler: the finished solution to #13 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 #13?
Puzzle #13 is rated hard. Its smallest color region holds 1 square, 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 #13 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #13, 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 #13?
Start with the teal region — the most constrained color on this board, with 1 square spanning rows 11–11 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.