12x12 Queens Puzzle #77

Difficulty: medium. 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 17-step reasoning is written out below.

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Step-by-step solution for puzzle #77

What follows is a complete solve of puzzle #77, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 12 queen placements and 5 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.

Show the full walkthrough (10 steps) — contains the solution

  1. Step 1. Only row 1, column 5 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 5, the rest of the indigo region and every square adjacent to it — diagonals included.

  2. Step 2. The rose region is down to one open square, at row 12, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 12 and column 9, the remaining rose squares, and the ring of neighbours around row 12, column 9.

  3. Step 3. Mint cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 10 squares in all.

    Puzzle #77 after step 3: 2 of 12 queens placed
    The board after step 3 — 2 of 12 queens placed.

  4. Step 4. Wherever amber ends up among its 5 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.

  5. Step 5. 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: 10 squares in all.

  6. Step 6. All of the blue region's remaining squares (2 open squares) lie in column 11. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 8 squares come off.

  7. Step 7. Only row 10, column 10 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 10, in sea green and in the ring around that square is now dead.

    Puzzle #77 after step 7: 3 of 12 queens placed
    The board after step 7 — 3 of 12 queens placed.

  8. Step 8. 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. Row 11 and column 12 are now spoken for, along with the squares touching that queen.

  9. Step 9. Amber and blue are now boxed into rows 2 and 3 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 3 squares at once.

  10. Step 10. 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: mint has one square left, so its queen takes row 4, column 1; slate has one square left, so its queen takes row 5, column 8; khaki has one square left, so its queen takes row 8, column 7; red has one square left, so its queen takes row 6, column 6; pale yellow has one square left, so its queen takes row 7, column 2; amber has one square left, so its queen takes row 2, column 3; blue has one square left, so its queen takes row 3, column 11; orange has one square left, so its queen takes row 9, 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 9, column 4.

    Puzzle #77 after step 10: 12 of 12 queens placed
    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 #77

Puzzle #77 is one of the medium-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 #77, 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 #77 are: indigo (1 square), rose (1 square), mint (2 squares), teal (2 squares), blue (8 squares), red (8 squares), khaki (12 squares), sea green (16 squares), pale yellow (17 squares), amber (18 squares), slate (29 squares), orange (30 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the orange region is the roomiest area on the grid with 30 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 orange with 30. A 29-square gap like this one rewards players who start where the choices are fewest.

Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #77, packing 100% of its 1×1 bounding box, whereas khaki straggles across its 7×4 box at just 43%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

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 occupies 1 square within rows 1–1, columns 5–5. It never leaves row 1, so its queen is locked to that row.

Rose: 1 square across rows 12–12, columns 9–9. It never leaves row 12, so its queen is locked to that row.

Mint covers 2 squares, spanning 2 rows and 1 column (rows 3–4, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Teal occupies 2 squares within rows 10–11, columns 12–12. It sits wholly inside column 12, so its queen must be in that column.

Blue: 8 squares across rows 1–5, columns 11–12.

Red covers 8 squares, spanning 3 rows and 4 columns (rows 5–7, columns 4–7).

Khaki occupies 12 squares within rows 6–12, columns 5–8.

Sea green: 16 squares across rows 7–12, columns 9–12.

Pale yellow covers 17 squares, spanning 6 rows and 4 columns (rows 3–8, columns 1–4).

Amber occupies 18 squares within rows 1–4, columns 1–8.

Slate: 29 squares across rows 1–9, columns 6–12.

Orange covers 30 squares, spanning 6 rows and 7 columns (rows 7–12, columns 1–7).

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 5 and 5. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #77.

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. Rose, mint 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 #77

A quick way to find the next move is to count how many different colors touch each line. On this board, row 2 is the narrowest, touched by only 3 colors. Column 10 is the tightest column, with 2 colors.

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 7, meeting 6 colors, is the opposite case — the busiest line on #77 and the last one you should expect to crack.

Working through the middle game

Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.

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 #77 is rated medium

Ratings track how generous the opening is. On #77 you get six regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 29-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.

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 medium board from the rest.

Stuck on #77 despite the medium 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.

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 #77, start hunting for it near column 10.

Here is a shape check that gives nothing away: when #77 is solved, 4 queens sit on the outer edge of the grid and 8 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.

How to play Queens

  1. The board is an N×N grid divided into N colored regions.
  2. Place exactly one queen (♛) in each row, each column, and each colored region.
  3. No two queens may touch each other — not horizontally, vertically, or diagonally.
  4. 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.
  5. 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 #77?

Puzzle #77 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and six of its regions are confined to a single row or column, which gives you a forced move early.

Does 12x12 puzzle #77 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #77, 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 #77?

Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.