12x12 Queens Puzzle #296

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 18-step reasoning is written out below.

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Step-by-step solution for puzzle #296

What follows is a complete solve of puzzle #296, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 12 queen placements and 6 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 line confined to one region.

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

Show the full walkthrough (8 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: mint has one square left, so its queen takes row 4, column 1; blue has one square left, so its queen takes row 3, column 4; teal has one square left, so its queen takes row 11, column 12; amber has one square left, so its queen takes row 2, column 11; indigo has one square left, so its queen takes row 1, column 6; red has one square left, so its queen takes row 6, column 2; rose has one square left, so its queen takes row 12, column 5. 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 5.

  2. Step 2. Pale yellow is down to 5 candidates in rows 7–9, columns 3, 7, and 8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.

  3. Step 3. Orange cannot leave column 3 — every one of its open squares (2 open squares) sits there. That column belongs to orange, so every square in it owned by another colour is dead: 1 square in all.

  4. Step 4. All of the sea green region's remaining squares (2 open squares) lie in column 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.

    Puzzle #296 after step 4: 7 of 12 queens placed
    The board after step 4 — 7 of 12 queens placed.

  5. Step 5. Khaki is down to 4 candidates in rows 8–10, columns 7–9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.

  6. Step 6. The pale yellow region has 3 squares left, spread across rows 7 and 8, columns 7 and 8. 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. Row 7 has only pale yellow squares left (2 open squares). The row needs a queen and only pale yellow can provide one, which pins that colour to this row and kills its squares elsewhere: 1 square.

  8. Step 8. 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: row 8 is down to one square, putting a queen on row 8, column 9; pale yellow has one square left, so its queen takes row 7, column 7; slate has one square left, so its queen takes row 5, column 8; sea green has one square left, so its queen takes row 10, column 10; orange has one square left, so its queen takes row 9, 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 9, column 3.

    Puzzle #296 after step 8: 12 of 12 queens placed
    The board after step 8 — 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 #296

This is puzzle number 296 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.

Puzzle #296 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 #296 are: mint (1 square), teal (1 square), blue (3 squares), indigo (6 squares), khaki (9 squares), rose (9 squares), amber (11 squares), sea green (16 squares), red (18 squares), pale yellow (18 squares), orange (23 squares), slate (29 squares). An even split would hand every color 12 squares, so the mint region is running 11 squares under that average while the slate region is the roomiest area on the grid with 29 squares.

Every color takes one queen and one only, whatever its size — which is why the 28 square difference between mint and slate 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 mint region is the most compact color here, filling 100% of the 1×1 box that encloses it, while rose is the most scattered at 45% of its 4×5 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

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.

Mint occupies 1 square within rows 4–4, columns 1–1. It never leaves row 4, so its queen is locked to that row.

Teal: 1 square across rows 11–11, columns 12–12. It never leaves row 11, so its queen is locked to that row.

Blue covers 3 squares, spanning 2 rows and 2 columns (rows 3–4, columns 4–5).

Indigo occupies 6 squares within rows 1–4, columns 5–6.

Khaki: 9 squares across rows 8–10, columns 7–11.

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

Amber occupies 11 squares within rows 1–4, columns 8–12.

Sea green: 16 squares across rows 8–12, columns 7–12.

Red covers 18 squares, spanning 6 rows and 5 columns (rows 1–6, columns 1–5).

Pale yellow occupies 18 squares within rows 5–9, columns 3–8.

Orange: 23 squares across rows 7–12, columns 1–5.

Slate covers 29 squares, spanning 7 rows and 7 columns (rows 1–7, columns 6–12).

Where to start on this board

The most constrained color is where a solve should begin, and on #296 that is mint with 1 square. It reaches from row 4 to row 4 and from column 1 to column 1 — 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 mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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. Teal and blue 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 #296

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 5, 6, 7 and 12 are the narrowest, touched by only 3 colors. Column 2 is the tightest column, with 2 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 4, meeting 6 colors, is the opposite case — the busiest line on #296 and the last one you should expect to crack.

Working through the middle game

The middle of #296 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.

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 #296 is rated medium

Ratings track how generous the opening is. On #296 you get four regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 28-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.

Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.

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 #296 is column 2.

Here is a shape check that gives nothing away: when #296 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 #296?

Puzzle #296 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.

Does 12x12 puzzle #296 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #296, 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 #296?

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