12x12 Queens Puzzle #78
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 22-step reasoning is written out below.
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Step-by-step solution for puzzle #78
What follows is a complete solve of puzzle #78, 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 proof by contradiction.
Techniques this board needs: last square in a region, region shadow, region confined to one line, proof by contradiction, pairing argument on columns, last square in a column.
Show the full walkthrough (14 steps) — contains the solution
Step 1. The board opens by cascading. Each of these placements is forced the moment the one before it is marked, so the whole run is really a single sweep repeated: indigo has one square left, so its queen takes row 1, column 1; red has one square left, so its queen takes row 6, column 4; pale yellow has one square left, so its queen takes row 7, column 7; sea green has one square left, so its queen takes row 10, column 10. 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 10, column 10.
Step 2. Amber is down to 5 candidates in rows 2–4, columns 9, 11, and 12, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 3. The slate region has 3 squares left, spread across rows 3 and 5, columns 2 and 3. 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 4. Teal cannot leave column 12 — every one of its open squares (2 open squares) sits there. That column belongs to teal, so every square in it owned by another colour is dead: 7 squares in all.
Step 5. Amber is down to 2 candidates in rows 2 and 3, columns 9 and 11, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about amber itself.
The board after step 5 — 4 of 12 queens placed. Step 6. 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: 6 squares in all.
Step 7. The teal region is down to one open square, at row 11, column 12. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 12, the remaining teal squares, and the ring of neighbours around row 11, column 12.
Step 8. 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: 3 squares in all.
Step 9. All of the khaki region's remaining squares (3 open squares) lie in row 8. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
The board after step 9 — 5 of 12 queens placed. Step 10. Test row 3, column 9 for amber: place a queen there, run the forced consequences through, and the blue region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 11. The amber region is down to one open square, at row 2, column 11. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 11, the remaining amber squares, and the ring of neighbours around row 2, column 11.
Step 12. Blue, mint, orange, and rose are now boxed into columns 5, 6, 8, and 9 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 1 square at once.
Step 13. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 5. Follow the forced moves that would create and row 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 6 queens fall out in order without any new idea being needed: column 5 is down to one square, putting a queen on row 4, column 5; blue has one square left, so its queen takes row 3, column 8; slate has one square left, so its queen takes row 5, column 2; khaki has one square left, so its queen takes row 8, column 3; orange has one square left, so its queen takes row 9, column 6; rose has one square left, so its queen takes row 12, column 9. 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 9.
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 #78
Board #78 of the 12x12 bank carries a easy 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, #78 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 #78 are: indigo (1 square), red (1 square), sea green (1 square), pale yellow (2 squares), teal (4 squares), slate (7 squares), rose (8 squares), amber (9 squares), mint (22 squares), orange (24 squares), blue (32 squares), khaki (33 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the khaki region is the roomiest area on the grid with 33 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 indigo, with 1 square, is under far more pressure than khaki with 33. A 32-square gap like this one rewards players who start where the choices are fewest.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and orange the most diffuse at 44% of its 6×9 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #78 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Indigo: 1 square across rows 1–1, columns 1–1. It never leaves row 1, so its queen is locked to that row.
Red covers 1 square, spanning 1 row and 1 column (rows 6–6, columns 4–4). It never leaves row 6, so its queen is locked to that row.
Sea green occupies 1 square within rows 10–10, columns 10–10. It never leaves row 10, so its queen is locked to that row.
Pale yellow: 2 squares across rows 6–7, columns 7–7. It sits wholly inside column 7, so its queen must be in that column.
Teal covers 4 squares, spanning 3 rows and 2 columns (rows 10–12, columns 11–12).
Slate occupies 7 squares within rows 2–5, columns 1–3.
Rose: 8 squares across rows 11–12, columns 7–11.
Amber covers 9 squares, spanning 4 rows and 4 columns (rows 1–4, columns 9–12).
Mint occupies 22 squares within rows 1–6, columns 2–9.
Orange: 24 squares across rows 7–12, columns 4–12.
Blue covers 32 squares, spanning 8 rows and 8 columns (rows 1–8, columns 5–12).
Khaki occupies 33 squares within rows 4–12, columns 1–6.
Where to start on this board
Indigo is the natural opening on #78: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 1–1. 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 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. Red, sea green, pale yellow 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 #78
A quick way to find the next move is to count how many different colors touch each line. On this board, row 9 is the narrowest, touched by only 2 colors. Columns 1, 2 and 3 are the tightest columns, with 3 colors.
A line with few colors is a line close to resolving. Where only 2 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.
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.
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 #78 is rated easy
What earns a board its rating is the shape of its opening. Here that opening offers seven single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 32. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a easy board from the rest.
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 #78, start hunting for it near row 9.
A non-spoiling distribution check: solved, #78 carries 3 queens on the perimeter against 9 queens inside, 1 of them cornered. 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 #78?
Puzzle #78 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and seven of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #78 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #78, 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 #78?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.