12x12 Queens Puzzle #45
Difficulty: easy. 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 26-step reasoning is written out below.
Previous puzzle · All 12x12 puzzles · Next puzzle
Step-by-step solution for puzzle #45
What follows is a complete solve of puzzle #45, worked the way a person works it rather than the way a computer does. It runs to 26 deductions — 12 queen placements and 14 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, line confined to one region.
Show the full walkthrough (19 steps) — contains the solution
Step 1. Only row 1, column 12 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 12, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. Blue is down to 3 candidates in rows 3 and 4, columns 10 and 11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 3. Orange cannot leave row 9 — every one of its open squares (2 open squares) sits there. That row belongs to orange, so every square on it owned by another colour is dead: 13 squares in all.
Step 4. Wherever sea green 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. Teal is down to 2 candidates in rows 10 and 11, columns 8 and 9, and they overlap enough to cover 4 squares elsewhere. Cross those out without deciding anything about teal itself.
The board after step 5 — 1 of 12 queens placed. Step 6. The orange region is down to one open square, at row 9, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 7, the remaining orange squares, and the ring of neighbours around row 9, column 7.
Step 7. Sea green cannot leave column 11 — every one of its open squares (2 open squares) sits there. That column belongs to sea green, so every square in it owned by another colour is dead: 6 squares in all.
Step 8. The blue region is down to one open square, at row 3, column 10. Every colour holds exactly one queen, so that square is forced. Row 3 and column 10 are now spoken for, along with the squares touching that queen.
Step 9. Sea green and teal are now boxed into rows 10 and 11 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 11 squares at once.
Step 10. All of the rose region's remaining squares (3 open squares) lie in row 12. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.
The board after step 10 — 3 of 12 queens placed. Step 11. Only row 8, column 1 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 1, in khaki and in the ring around that square is now dead.
Step 12. The pale yellow region has 4 squares left, spread across rows 6 and 7, columns 3, 5, and 6. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 13. Wherever red 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 14. Here is the pairing argument. The red and teal regions have no open squares outside columns 8 and 9 — 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: 4 squares come off in one stroke.
The board after step 14 — 4 of 12 queens placed. Step 15. 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: 8 squares in all.
Step 16. All of the pale yellow region's remaining squares (2 open squares) lie in row 7. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 1 square come off.
Step 17. Row 2 has only amber squares left (5 open squares). The row needs a queen and only amber can provide one, which pins that colour to this row and kills its squares elsewhere: 2 squares.
Step 18. Here is the pairing argument. The slate and rose regions have no open squares outside columns 5 and 6 — 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 19. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: pale yellow has one square left, so its queen takes row 7, column 3; mint has one square left, so its queen takes row 4, column 4; amber has one square left, so its queen takes row 2, column 2; slate has one square left, so its queen takes row 5, column 6; red has one square left, so its queen takes row 6, column 9; teal has one square left, so its queen takes row 11, column 8; sea green has one square left, so its queen takes row 10, column 11; rose has one square left, so its queen takes row 12, column 5. 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 5.
The board after step 19 — 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 #45
Puzzle #45 is one of the easy-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 #45, 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 #45 are: indigo (1 square), orange (2 squares), teal (6 squares), mint (8 squares), sea green (8 squares), blue (9 squares), khaki (11 squares), slate (12 squares), rose (16 squares), pale yellow (17 squares), red (25 squares), amber (29 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the amber 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 28 square difference between indigo and amber 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.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; slate is the loosest, occupying only 40% of its 6×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Indigo: 1 square across rows 1–1, columns 12–12. It never leaves row 1, so its queen is locked to that row.
Orange covers 2 squares, spanning 1 row and 2 columns (rows 9–9, columns 7–8). It never leaves row 9, so its queen is locked to that row.
Teal occupies 6 squares within rows 9–11, columns 8–10.
Mint: 8 squares across rows 4–7, columns 2–5.
Sea green covers 8 squares, spanning 4 rows and 3 columns (rows 9–12, columns 10–12).
Blue occupies 9 squares within rows 1–4, columns 10–12.
Khaki: 11 squares across rows 8–12, columns 1–4.
Slate covers 12 squares, spanning 6 rows and 5 columns (rows 2–7, columns 5–9).
Rose occupies 16 squares within rows 9–12, columns 3–11.
Pale yellow: 17 squares across rows 6–10, columns 1–6.
Red covers 25 squares, spanning 6 rows and 7 columns (rows 3–8, columns 6–12).
Amber occupies 29 squares within rows 1–5, columns 1–10.
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 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 #45.
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. Orange is no more than two rows tall, so its queen is 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 #45
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 8 and 12 are the narrowest, touched by only 3 colors. Column 1 is the tightest column, 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 9 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
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.
Two checks, run alternately, will carry you through the 144 squares of this grid. First: has any of the twelve regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 12x12 boards almost always run one of those checks and forget the other.
A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.
Why puzzle #45 is rated easy
Ratings track how generous the opening is. On #45 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 28-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.
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 #45 despite the easy 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 #45 is row 1.
As a sanity check rather than a spoiler: the finished solution to #45 puts 3 queens on the outer border and 9 queens in the interior, 1 of them in an actual 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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 12x12 Queens puzzle #45?
Puzzle #45 is rated easy. Its smallest color region holds 1 square, 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 #45 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #45, 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 #45?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.