12x12 Queens Puzzle #58

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 17-step reasoning is written out below.

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Step-by-step solution for puzzle #58

What follows is a complete solve of puzzle #58, 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 region shadow.

Techniques this board needs: last square in a region, region confined to one line, region shadow, last square in a row.

Show the full walkthrough (9 steps) — contains the solution

  1. 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 4; amber has one square left, so its queen takes row 2, column 11; mint has one square left, so its queen takes row 4, column 10; slate has one square left, so its queen takes row 5, column 12; blue has one square left, so its queen takes row 3, column 3. 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 3, column 3.

  2. Step 2. All of the red region's remaining squares (3 open squares) lie in row 6. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.

  3. Step 3. Only row 7, column 9 is still open in pale yellow, and since each colour must take a queen, that settles it. Everything else in row 7, in column 9, in pale yellow and in the ring around that square is now dead.

    Puzzle #58 after step 3: 6 of 12 queens placed
    The board after step 3 — 6 of 12 queens placed.

  4. Step 4. The teal region is down to one open square, at row 11, column 8. Every colour holds exactly one queen, so that square is forced. Row 11 and column 8 are now spoken for, along with the squares touching that queen.

  5. Step 5. The khaki region has 3 squares left, spread across rows 8 and 10, columns 5–7. 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.

  6. Step 6. Wherever orange ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

    Puzzle #58 after step 6: 7 of 12 queens placed
    The board after step 6 — 7 of 12 queens placed.

  7. Step 7. Sea green is down to 4 candidates in rows 9, 10, and 12, columns 5–7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about sea green itself.

  8. Step 8. All of the khaki region's remaining squares (2 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.

  9. Step 9. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: orange has one square left, so its queen takes row 9, column 1; row 10 is down to one square, putting a queen on row 10, column 5; 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; rose has one square left, so its queen takes row 12, column 2. 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 2.

    Puzzle #58 after step 9: 12 of 12 queens placed
    The board after step 9 — 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 #58

Board #58 of the 12x12 bank carries a hard 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, #58 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 #58 are: indigo (1 square), amber (4 squares), slate (5 squares), mint (8 squares), rose (11 squares), orange (12 squares), pale yellow (13 squares), sea green (15 squares), teal (15 squares), blue (17 squares), khaki (19 squares), red (24 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the red region is the roomiest area on the grid with 24 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 red with 24. A 23-square gap like this one rewards players who start where the choices are fewest.

Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; khaki is the loosest, occupying only 34% of its 8×7 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 covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 4–4). It never leaves row 1, so its queen is locked to that row.

Amber occupies 4 squares within rows 1–2, columns 9–11.

Slate: 5 squares across rows 4–6, columns 10–12.

Mint covers 8 squares, spanning 4 rows and 3 columns (rows 1–4, columns 10–12).

Rose occupies 11 squares within rows 9–12, columns 1–5.

Orange: 12 squares across rows 3–9, columns 1–2.

Pale yellow covers 13 squares, spanning 6 rows and 4 columns (rows 4–9, columns 9–12).

Sea green occupies 15 squares within rows 8–12, columns 3–8.

Teal: 15 squares across rows 8–12, columns 8–12.

Blue covers 17 squares, spanning 5 rows and 5 columns (rows 1–5, columns 1–5).

Khaki occupies 19 squares within rows 4–11, columns 3–9.

Red: 24 squares across rows 1–7, columns 3–9.

Where to start on this board

Indigo is the natural opening on #58: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 4–4. 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. Amber 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 #58

A quick way to find the next move is to count how many different colors touch each line. On this board, row 12 is the narrowest, touched by only 3 colors. Columns 1, 2, 6 and 7 are the tightest columns, with 3 colors.

Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 11 rows. Row 4 sits at the far end with 7 colors crossing it; it will almost certainly be among the last to settle.

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.

Keep alternating between two views of the board. In the region view you ask which of the twelve colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 12x12 board unwinds.

Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.

Why puzzle #58 is rated hard

Ratings track how generous the opening is. On #58 you get two regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 23-square gap between the largest and smallest regions. Those four numbers largely explain the hard 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.

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 #58, start hunting for it near row 12.

Use this as a rough check on your progress: the completed board has 4 queens around the border and 8 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.

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 #58?

Puzzle #58 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 #58 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #58, 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 #58?

Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.