12x12 Queens Puzzle #56

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 17-step reasoning is written out below.

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Step-by-step solution for puzzle #56

What follows is a complete solve of puzzle #56, 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 columns.

Techniques this board needs: last square in a region, region shadow, region confined to one line, pairing argument on columns.

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 5; pale yellow has one square left, so its queen takes row 7, column 12; rose has one square left, so its queen takes row 12, column 2; red has one square left, so its queen takes row 6, column 3; slate has one square left, so its queen takes row 5, column 11. 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 5, column 11.

  2. Step 2. The amber region has 5 squares left, spread across rows 2–4, columns 7, 9, and 10. 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.

  3. Step 3. Khaki cannot leave row 8 — every one of its open squares (2 open squares) sits there. That row belongs to khaki, so every square on it owned by another colour is dead: 7 squares in all.

    Puzzle #56 after step 3: 5 of 12 queens placed
    The board after step 3 — 5 of 12 queens placed.

  4. Step 4. Sea green is down to 2 candidates in rows 10 and 11, columns 1 and 6, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about sea green itself.

  5. Step 5. Teal cannot leave row 11 — every one of its open squares (2 open squares) sits there. That row belongs to teal, so every square on it owned by another colour is dead: 5 squares in all.

  6. Step 6. The sea green region is down to one open square, at row 10, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 1, the remaining sea green squares, and the ring of neighbours around row 10, column 1.

    Puzzle #56 after step 6: 6 of 12 queens placed
    The board after step 6 — 6 of 12 queens placed.

  7. Step 7. Only row 3, column 4 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 4, in blue and in the ring around that square is now dead.

  8. Step 8. Here is the pairing argument. The khaki and teal regions have no open squares outside columns 7 and 8 — 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.

  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: mint has one square left, so its queen takes row 4, column 6; amber has one square left, so its queen takes row 2, column 10; orange has one square left, so its queen takes row 9, column 9; khaki has one square left, so its queen takes row 8, column 7; teal has one square left, so its queen takes row 11, 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 11, column 8.

    Puzzle #56 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 #56

This is puzzle number 56 in the 12x12 collection, rated easy. The grid holds 144 squares split between twelve colored regions, and a finished board carries twelve queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

Puzzle #56 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.

How the colors divide this board

From smallest to largest, the regions on puzzle #56 are: indigo (1 square), pale yellow (1 square), rose (1 square), red (5 squares), khaki (5 squares), slate (8 squares), teal (8 squares), sea green (16 squares), amber (20 squares), mint (23 squares), blue (26 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.

Every color takes one queen and one only, whatever its size — which is why the 29 square difference between indigo and orange 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.

Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×1 box that encloses it, while teal is the most scattered at 53% of its 3×5 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.

Region by region

Here is the full footprint of every color on puzzle #56, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

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

Pale yellow: 1 square across rows 7–7, columns 12–12. It never leaves row 7, so its queen is locked to that row.

Rose covers 1 square, spanning 1 row and 1 column (rows 12–12, columns 2–2). It never leaves row 12, so its queen is locked to that row.

Red occupies 5 squares within rows 6–7, columns 2–4.

Khaki: 5 squares across rows 7–8, columns 5–8.

Slate covers 8 squares, spanning 3 rows and 4 columns (rows 5–7, columns 9–12).

Teal occupies 8 squares within rows 10–12, columns 6–10.

Sea green: 16 squares across rows 8–12, columns 1–6.

Amber covers 20 squares, spanning 5 rows and 7 columns (rows 1–5, columns 6–12).

Mint occupies 23 squares within rows 2–8, columns 4–9.

Blue: 26 squares across rows 1–9, columns 1–5.

Orange covers 30 squares, spanning 5 rows and 10 columns (rows 8–12, columns 3–12).

Where to start on this board

The most constrained color is where a solve should begin, and on #56 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 5 to column 5 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on 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. Pale yellow, rose, red and khaki 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 #56

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3, 4, 9, 10 and 11 are the narrowest, touched by only 3 colors. Column 1 is the tightest column, with 2 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 7 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.

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.

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 #56 is rated easy

The rating reflects how much the board concedes up front. Puzzle #56 has six regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 29 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.

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 #56 is column 1.

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 #56?

Puzzle #56 is rated easy. 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 #56 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #56, 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 #56?

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.