12x12 Queens Puzzle #174

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 20-step reasoning is written out below.

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Step-by-step solution for puzzle #174

What follows is a complete solve of puzzle #174, 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 shadow, region confined to one line, pairing argument on rows, pairing argument on columns, last square in a row.

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

  1. Step 1. Only row 4, column 7 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 7, the rest of the mint region and every square adjacent to it — diagonals included.

  2. Step 2. Indigo is down to 5 candidates in rows 1–3, columns 10–12, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.

  3. Step 3. Red cannot leave row 6 — every one of its open squares (2 open squares) sits there. That row belongs to red, so every square on it owned by another colour is dead: 13 squares in all.

  4. Step 4. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: pale yellow has one square left, so its queen takes row 7, column 12; orange has one square left, so its queen takes row 9, column 11; indigo has one square left, so its queen takes row 1, column 10; rose has one square left, so its queen takes row 12, column 9. 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 9.

    Puzzle #174 after step 4: 5 of 12 queens placed
    The board after step 4 — 5 of 12 queens placed.

  5. Step 5. Amber is down to 3 candidates in rows 2 and 5, columns 5, 6, and 8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.

  6. Step 6. The blue region has 4 squares left, spread across rows 2 and 3, columns 1–3. 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. Wherever teal 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.

    Puzzle #174 after step 7: 5 of 12 queens placed
    The board after step 7 — 5 of 12 queens placed.

  8. Step 8. Khaki, sea green, and teal are now boxed into rows 8, 10, and 11 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 2 squares at once.

  9. Step 9. Here is the pairing argument. The blue, red, and khaki regions have no open squares outside columns 1–3 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 3 squares come off in one stroke.

  10. Step 10. Slate and teal are now boxed into columns 4 and 5 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 1 square at once.

  11. Step 11. 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; teal has one square left, so its queen takes row 11, column 5; row 3 is down to one square, putting a queen on row 3, column 1; red has one square left, so its queen takes row 6, column 2; khaki has one square left, so its queen takes row 8, column 3; sea green has one square left, so its queen takes row 10, column 8; amber has one square left, so its queen takes row 2, 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 2, column 6.

    Puzzle #174 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 #174

Board #174 of the 12x12 bank carries a easy rating. Its 144 squares are shared between twelve colored regions, and the solution places twelve queens: one per row, one per column, one per region, none of them side by side or corner to corner.

As with every puzzle in this collection, #174 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 #174 are: mint (1 square), red (2 squares), pale yellow (3 squares), indigo (5 squares), rose (6 squares), blue (8 squares), orange (9 squares), teal (9 squares), khaki (20 squares), sea green (21 squares), slate (24 squares), amber (36 squares). An even split would hand every color 12 squares, so the mint region is running 11 squares under that average while the amber region is the roomiest area on the grid with 36 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Mint has only 1 place to put its queen; amber has 36. The spread between them on this board is 35 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Look at arrangement as well as area. Mint is the densest region on this grid at 100% of a 1×1 frame; orange is the loosest, occupying only 45% of its 5×4 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

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 easy board.

Mint: 1 square across rows 4–4, columns 7–7. It never leaves row 4, so its queen is locked to that row.

Red covers 2 squares, spanning 1 row and 2 columns (rows 6–6, columns 1–2). It never leaves row 6, so its queen is locked to that row.

Pale yellow occupies 3 squares within rows 6–7, columns 11–12.

Indigo: 5 squares across rows 1–3, columns 10–12.

Rose covers 6 squares, spanning 2 rows and 4 columns (rows 11–12, columns 9–12).

Blue occupies 8 squares within rows 1–3, columns 1–4.

Orange: 9 squares across rows 7–11, columns 9–12.

Teal covers 9 squares, spanning 3 rows and 5 columns (rows 10–12, columns 3–7).

Khaki occupies 20 squares within rows 7–12, columns 1–5.

Sea green: 21 squares across rows 7–12, columns 6–11.

Slate covers 24 squares, spanning 7 rows and 7 columns (rows 2–8, columns 1–7).

Amber occupies 36 squares within rows 1–6, columns 5–12.

Where to start on this board

Mint is the natural opening on #174: it is the smallest region on the grid at 1 square, boxed into rows 4–4 and columns 7–7. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of 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. Red, 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 #174

A quick way to find the next move is to count how many different colors touch each line. On this board, row 5 is the narrowest, touched by only 2 colors. Column 8 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 2 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

The middle of #174 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.

Keep alternating between two views of the board. In the region view you ask which of the twelve colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 12x12 board unwinds.

The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.

Why puzzle #174 is rated easy

The easy label comes out of four measurable things about #174: three of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 35. Boards that score tight on those measures give up their first queen fast.

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.

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.

One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #174 the place to look for one is around row 5.

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 #174?

Puzzle #174 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.

Does 12x12 puzzle #174 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #174, 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 #174?

Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.