12x12 Queens Puzzle #28

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 23-step reasoning is written out below.

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Step-by-step solution for puzzle #28

What follows is a complete solve of puzzle #28, worked the way a person works it rather than the way a computer does. It runs to 23 deductions — 12 queen placements and 11 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, pairing argument on rows, pairing argument on columns, region confined to one line.

Show the full walkthrough (16 steps) — contains the solution

  1. Step 1. Only row 2, column 12 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 12, the rest of the amber region and every square adjacent to it — diagonals included.

  2. Step 2. The indigo region is down to one open square, at row 1, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 4, the remaining indigo squares, and the ring of neighbours around row 1, column 4.

  3. Step 3. The pale yellow region has 3 squares left, spread across rows 7 and 8, columns 1 and 2. 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.

  4. Step 4. Wherever sea green 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.

  5. Step 5. Sea green, teal, and rose are now boxed into rows 10–12 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 12 squares at once.

    Puzzle #28 after step 5: 2 of 12 queens placed
    The board after step 5 — 2 of 12 queens placed.

  6. Step 6. Here is the pairing argument. The pale yellow and sea green regions have no open squares outside columns 1 and 2 — 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: 10 squares come off in one stroke.

  7. Step 7. Slate cannot leave column 3 — every one of its open squares (3 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 6 squares in all.

  8. Step 8. The rose region is down to one open square, at row 12, column 5. Every colour holds exactly one queen, so that square is forced. Row 12 and column 5 are now spoken for, along with the squares touching that queen.

  9. Step 9. The blue region has 3 squares left, spread across rows 3 and 4, columns 6 and 7. 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.

  10. Step 10. Wherever red ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.

  11. Step 11. Khaki is down to 4 candidates in rows 8 and 9, columns 6–8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.

    Puzzle #28 after step 11: 3 of 12 queens placed
    The board after step 11 — 3 of 12 queens placed.

  12. Step 12. The teal region has 3 squares left, spread across rows 10 and 11, columns 8 and 9. 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.

  13. Step 13. Blue and mint are now boxed into rows 3 and 4 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.

  14. Step 14. The slate region is down to one open square, at row 5, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 3, the remaining slate squares, and the ring of neighbours around row 5, column 3.

  15. Step 15. Blue and red are now boxed into columns 6 and 7 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 3 squares at once.

  16. Step 16. 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: khaki has one square left, so its queen takes row 8, column 8; pale yellow has one square left, so its queen takes row 7, column 1; red has one square left, so its queen takes row 6, column 6; blue has one square left, so its queen takes row 3, column 7; sea green has one square left, so its queen takes row 10, column 2; teal has one square left, so its queen takes row 11, column 9; mint has one square left, so its queen takes row 4, column 11; orange has one square left, so its queen takes row 9, 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 9, column 10.

    Puzzle #28 after step 16: 12 of 12 queens placed
    The board after step 16 — 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 #28

This is puzzle number 28 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.

There is exactly one way to finish #28, 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 #28 are: amber (1 square), indigo (2 squares), pale yellow (3 squares), sea green (5 squares), rose (7 squares), khaki (11 squares), teal (12 squares), blue (15 squares), mint (18 squares), slate (18 squares), red (22 squares), orange (30 squares). An even split would hand every color 12 squares, so the amber region is running 11 squares under that average while the orange region is the roomiest area on the grid with 30 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 orange has 30 to spare, a gap of 29. Attack the cramped end and the roomy regions tend to resolve themselves.

Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 2×1 box that encloses it, while teal is the most scattered at 57% of its 3×7 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 #28, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

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

Indigo occupies 2 squares within rows 1–2, columns 4–4. It sits wholly inside column 4, so its queen must be in that column.

Pale yellow: 3 squares across rows 7–8, columns 1–2.

Sea green covers 5 squares, spanning 4 rows and 2 columns (rows 9–12, columns 1–2).

Rose occupies 7 squares within rows 11–12, columns 2–5.

Khaki: 11 squares across rows 8–11, columns 5–8.

Teal covers 12 squares, spanning 3 rows and 7 columns (rows 10–12, columns 6–12).

Blue occupies 15 squares within rows 1–5, columns 4–8.

Mint: 18 squares across rows 1–5, columns 7–12.

Slate covers 18 squares, spanning 6 rows and 4 columns (rows 1–6, columns 1–4).

Red occupies 22 squares within rows 5–10, columns 2–7.

Orange: 30 squares across rows 4–11, columns 7–12.

Where to start on this board

The most constrained color is where a solve should begin, and on #28 that is amber with 1 square. It reaches from row 2 to row 2 and from column 12 to column 12 — 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. Indigo, 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 #28

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3, 6, 7 and 12 are the narrowest, touched by only 3 colors. Columns 1, 3, 9, 10 and 11 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 11 sits at the far end with 5 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.

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 #28 is rated easy

The rating reflects how much the board concedes up front. Puzzle #28 has three 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.

Stuck on #28 despite the easy 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.

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 3 on this board.

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 #28?

Puzzle #28 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.

Does 12x12 puzzle #28 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #28, 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 #28?

Start with the amber region — the most constrained color on this board, with 1 square spanning rows 2–2 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.