12x12 Queens Puzzle #52

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 20-step reasoning is written out below.

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Step-by-step solution for puzzle #52

What follows is a complete solve of puzzle #52, 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 rows.

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

Show the full walkthrough (11 steps) — contains the solution

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

  2. Step 2. Orange is down to 6 candidates in rows 7–11, columns 1 and 2, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about orange itself.

  3. Step 3. Only row 11, column 1 is still open in teal, and since each colour must take a queen, that settles it. Everything else in row 11, in column 1, in teal and in the ring around that square is now dead.

  4. Step 4. The orange region is down to one open square, at row 9, column 2. Every colour holds exactly one queen, so that square is forced. Row 9 and column 2 are now spoken for, along with the squares touching that queen.

    Puzzle #52 after step 4: 2 of 12 queens placed
    The board after step 4 — 2 of 12 queens placed.

  5. Step 5. Pale yellow is down to 4 candidates in rows 2, 6, and 7, columns 3 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.

  6. Step 6. The rose region has 2 squares left, spread across rows 8 and 12, columns 3 and 4. 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.

  7. Step 7. Indigo, amber, and blue are now boxed into rows 1–3 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 11 squares at once.

    Puzzle #52 after step 7: 2 of 12 queens placed
    The board after step 7 — 2 of 12 queens placed.

  8. Step 8. All of the mint region's remaining squares (3 open squares) lie in row 4. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 9 squares come off.

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

  10. Step 10. All of the slate region's remaining squares (3 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.

  11. Step 11. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: red has one square left, so its queen takes row 6, column 12; pale yellow has one square left, so its queen takes row 7, column 4; rose has one square left, so its queen takes row 12, column 3; blue has one square left, so its queen takes row 3, column 5; slate has one square left, so its queen takes row 5, column 6; row 2 is down to one square, putting a queen on row 2, column 10; khaki has one square left, so its queen takes row 8, column 7; indigo has one square left, so its queen takes row 1, column 8; mint has one square left, so its queen takes row 4, column 9; sea green has one square left, so its queen takes row 10, 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 10, column 11.

    Puzzle #52 after step 11: 12 of 12 queens placed
    The board after step 11 — 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 #52

This is puzzle number 52 in the 12x12 collection, rated hard. 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 #52, 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 #52 are: teal (2 squares), indigo (6 squares), amber (6 squares), orange (6 squares), mint (8 squares), blue (10 squares), rose (11 squares), slate (12 squares), red (12 squares), pale yellow (14 squares), sea green (24 squares), khaki (33 squares). An even split would hand every color 12 squares, so the teal region is running 10 squares under that average while the khaki region is the roomiest area on the grid with 33 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Teal is working with 2 squares while khaki has 33 to spare, a gap of 31. Attack the cramped end and the roomy regions tend to resolve themselves.

Shape matters as much as size. The teal region is the most compact color here, filling 100% of the 2×1 box that encloses it, while rose is the most scattered at 46% of its 6×4 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

Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a hard board.

Teal covers 2 squares, spanning 2 rows and 1 column (rows 10–11, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Indigo occupies 6 squares within rows 1–2, columns 5–9.

Amber: 6 squares across rows 1–3, columns 9–12.

Orange covers 6 squares, spanning 5 rows and 2 columns (rows 7–11, columns 1–2).

Mint occupies 8 squares within rows 2–4, columns 7–10.

Blue: 10 squares across rows 1–3, columns 1–5.

Rose covers 11 squares, spanning 6 rows and 4 columns (rows 7–12, columns 1–4).

Slate occupies 12 squares within rows 3–6, columns 2–7.

Red: 12 squares across rows 2–6, columns 8–12.

Pale yellow covers 14 squares, spanning 7 rows and 4 columns (rows 2–8, columns 1–4).

Sea green occupies 24 squares within rows 7–12, columns 7–12.

Khaki: 33 squares across rows 5–12, columns 4–11.

Where to start on this board

The most constrained color is where a solve should begin, and on #52 that is teal with 2 squares. It reaches from row 10 to row 11 and from column 1 to column 1 — 2 rows 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 teal stays inside column 1, its queen belongs to that column and no other region may use it. Cross out the rest of column 1 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.

Work outward to the other flat regions next. Indigo 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 #52

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 12 are the narrowest, touched by only 3 colors. Columns 6 and 12 are the tightest columns, with 3 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 3, meeting 6 colors, is the opposite case — the busiest line on #52 and the last one you should expect to crack.

Working through the middle game

The middle of #52 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.

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.

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 #52 is rated hard

The hard label comes out of four measurable things about #52: one of its regions is locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 31. Boards that score tight on those measures give up their first queen fast.

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.

Stuck on #52 despite the hard 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 1 on this board.

Here is a shape check that gives nothing away: when #52 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 #52?

Puzzle #52 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.

Does 12x12 puzzle #52 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #52, 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 #52?

Start with the teal region — the most constrained color on this board, with 2 squares spanning rows 10–11 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.