12x12 Queens Puzzle #104

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 20-step reasoning is written out below.

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Step-by-step solution for puzzle #104

What follows is a complete solve of puzzle #104, worked the way a person works it rather than the way a computer does. It runs to 20 deductions — 12 queen placements and 8 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 confined to one line, region shadow, pairing argument on rows, pairing argument on columns, last square in a row.

Show the full walkthrough (14 steps) — contains the solution

  1. Step 1. Only row 2, column 11 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 11, the rest of the amber region and every square adjacent to it — diagonals included.

  2. Step 2. The red region is down to one open square, at row 6, column 12. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 12, the remaining red squares, and the ring of neighbours around row 6, column 12.

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

  4. Step 4. All of the indigo region's remaining squares (2 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 7 squares come off.

  5. Step 5. Only row 7, column 2 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 2, the rest of the pale yellow region and every square adjacent to it — diagonals included.

    Puzzle #104 after step 5: 4 of 12 queens placed
    The board after step 5 — 4 of 12 queens placed.

  6. Step 6. All of the blue region's remaining squares (2 open squares) lie in column 3. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 7 squares come off.

  7. Step 7. Wherever khaki ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

  8. Step 8. Rose is down to 4 candidates in rows 11 and 12, columns 4–6, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about rose itself.

  9. Step 9. Indigo, blue, mint, and slate are now boxed into rows 1, 3, 4, and 5 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 4 squares at once.

    Puzzle #104 after step 9: 4 of 12 queens placed
    The board after step 9 — 4 of 12 queens placed.

  10. Step 10. The khaki region is down to one open square, at row 8, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 10, the remaining khaki squares, and the ring of neighbours around row 8, column 10.

  11. Step 11. Sea green cannot leave row 10 — every one of its open squares (2 open squares) sits there. That row belongs to sea green, so every square on it owned by another colour is dead: 3 squares in all.

  12. Step 12. All of the rose region's remaining squares (2 open squares) lie in row 12. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.

  13. Step 13. Slate, sea green, and rose are now boxed into columns 4–6 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 4 squares at once.

  14. Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: row 5 is down to one square, putting a queen on row 5, column 4; blue has one square left, so its queen takes row 3, column 3; indigo has one square left, so its queen takes row 1, column 1; mint has one square left, so its queen takes row 4, column 8; sea green has one square left, so its queen takes row 10, column 5; teal has one square left, so its queen takes row 11, column 9; rose has one square left, so its queen takes row 12, column 6. 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 6.

    Puzzle #104 after step 14: 12 of 12 queens placed
    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 #104

This is puzzle number 104 in the 12x12 collection, rated medium. 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.

As with every puzzle in this collection, #104 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 #104 are: amber (1 square), orange (1 square), indigo (3 squares), red (3 squares), blue (4 squares), rose (4 squares), pale yellow (7 squares), khaki (12 squares), slate (23 squares), teal (23 squares), mint (31 squares), sea green (32 squares). An even split would hand every color 12 squares, so the amber region is running 11 squares under that average while the sea green region is the roomiest area on the grid with 32 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 1 square while sea green has 32 to spare, a gap of 31. Attack the cramped end and the roomy regions tend to resolve themselves.

Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #104, packing 100% of its 3×1 bounding box, whereas sea green straggles across its 8×8 box at just 50%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Here is the full footprint of every color on puzzle #104, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Amber occupies 1 square within rows 2–2, columns 11–11. It never leaves row 2, so its queen is locked to that row.

Orange: 1 square across rows 9–9, columns 7–7. It never leaves row 9, so its queen is locked to that row.

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

Red occupies 3 squares within rows 5–6, columns 11–12.

Blue: 4 squares across rows 3–4, columns 1–3.

Rose covers 4 squares, spanning 2 rows and 3 columns (rows 11–12, columns 4–6).

Pale yellow occupies 7 squares within rows 5–8, columns 1–3.

Khaki: 12 squares across rows 4–9, columns 9–12.

Slate covers 23 squares, spanning 8 rows and 5 columns (rows 1–8, columns 2–6).

Teal occupies 23 squares within rows 7–12, columns 7–12.

Mint: 31 squares across rows 1–6, columns 5–12.

Sea green covers 32 squares, spanning 8 rows and 8 columns (rows 5–12, columns 1–8).

Where to start on this board

The most constrained color is where a solve should begin, and on #104 that is amber with 1 square. It reaches from row 2 to row 2 and from column 11 to column 11 — 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 amber never leaves row 2, its queen must be in that row — and that instantly settles row 2 for everyone else. Sweep along it and X out every square that is not amber. 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, blue, red 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 #104

A quick way to find the next move is to count how many different colors touch each line. On this board, row 10 is the narrowest, touched by only 2 colors. Columns 4, 8, 9 and 10 are the tightest columns, with 3 colors.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 11 rows. By contrast row 6 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

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.

With twelve queens to seat and twelve regions competing over 144 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 12x12 boards quickly and players who stall halfway.

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 #104 is rated medium

The rating reflects how much the board concedes up front. Puzzle #104 has five regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 31 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.

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.

Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 10 on this board.

As a sanity check rather than a spoiler: the finished solution to #104 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

  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 #104?

Puzzle #104 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.

Does 12x12 puzzle #104 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #104, 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 #104?

Start with the amber region — the most constrained color on this board, with 1 square spanning rows 2–2 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.