12x12 Queens Puzzle #109

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 21-step reasoning is written out below.

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Step-by-step solution for puzzle #109

What follows is a complete solve of puzzle #109, worked the way a person works it rather than the way a computer does. It runs to 21 deductions — 12 queen placements and 9 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 region shadow.

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

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

  1. Step 1. Only row 1, column 5 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 5, the rest of the indigo region and every square adjacent to it — diagonals included.

  2. Step 2. The pale yellow region is down to one open square, at row 7, column 12. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 12, the remaining pale yellow squares, and the ring of neighbours around row 7, column 12.

  3. Step 3. Only row 2, column 11 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 11, in amber and in the ring around that square is now dead.

  4. Step 4. Wherever red ends up among its 3 squares, the same squares fall within reach of it — 4 squares in total, impossible regardless of how the colour resolves.

  5. Step 5. Sea green cannot leave column 10 — every one of its open squares (2 open squares) sits there. That column belongs to sea green, so every square in it owned by another colour is dead: 8 squares in all.

    Puzzle #109 after step 5: 3 of 12 queens placed
    The board after step 5 — 3 of 12 queens placed.

  6. Step 6. 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: rose has one square left, so its queen takes row 12, column 9; sea green has one square left, so its queen takes row 10, column 10; teal has one square left, so its queen takes row 11, column 1; red has one square left, so its queen takes row 6, column 2. 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 6, column 2.

  7. Step 7. Wherever slate ends up among its 4 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.

  8. Step 8. Orange is down to 3 candidates in rows 8 and 9, columns 6 and 8, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about orange itself.

  9. Step 9. The khaki region has 3 squares left, spread across rows 5 and 8, 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.

    Puzzle #109 after step 9: 7 of 12 queens placed
    The board after step 9 — 7 of 12 queens placed.

  10. Step 10. Blue cannot leave row 3 — every one of its open squares (4 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 1 square in all.

  11. Step 11. Mint is down to 3 candidates in rows 4 and 5, columns 7 and 8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.

  12. Step 12. Khaki cannot leave column 6 — every one of its open squares (2 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 2 squares in all.

  13. Step 13. All of the orange region's remaining squares (2 open squares) lie in column 8. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.

  14. Step 14. 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: mint has one square left, so its queen takes row 4, column 7; khaki has one square left, so its queen takes row 8, column 6; orange has one square left, so its queen takes row 9, column 8; slate has one square left, so its queen takes row 5, column 4; blue has one square left, so its queen takes row 3, 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 3, column 3.

    Puzzle #109 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 #109

Puzzle #109 is one of the medium-rated boards in the 12x12 set. Twelve colors divide its 144 squares, and you win by placing twelve queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.

There is exactly one way to finish #109, 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 #109 are: indigo (1 square), pale yellow (1 square), rose (4 squares), red (5 squares), teal (6 squares), khaki (8 squares), amber (9 squares), sea green (11 squares), slate (16 squares), blue (24 squares), mint (24 squares), orange (35 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the orange region is the roomiest area on the grid with 35 squares.

Every color takes one queen and one only, whatever its size — which is why the 34 square difference between indigo and orange 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.

Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #109, packing 100% of its 1×1 bounding box, whereas amber straggles across its 6×3 box at just 50%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Below is every color on #109 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 5–5). It never leaves row 1, so its queen is locked to that row.

Pale yellow occupies 1 square within rows 7–7, columns 12–12. It never leaves row 7, so its queen is locked to that row.

Rose: 4 squares across rows 11–12, columns 9–11.

Red covers 5 squares, spanning 4 rows and 2 columns (rows 6–9, columns 1–2).

Teal occupies 6 squares within rows 8–12, columns 1–2.

Khaki: 8 squares across rows 5–9, columns 6–8.

Amber covers 9 squares, spanning 6 rows and 3 columns (rows 1–6, columns 10–12).

Sea green occupies 11 squares within rows 8–12, columns 10–12.

Slate: 16 squares across rows 4–9, columns 3–6.

Blue covers 24 squares, spanning 6 rows and 7 columns (rows 1–6, columns 1–7).

Mint occupies 24 squares within rows 1–7, columns 6–11.

Orange: 35 squares across rows 6–12, columns 2–10.

Where to start on this board

Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 5 and 5. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #109.

Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. 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 #109

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 3 and 10 are the narrowest, touched by only 3 colors. Columns 1, 3, 4, 8, 9 and 12 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 6 sits at the far end with 7 colors crossing it; it will almost certainly be among the last to settle.

Working through the middle game

After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.

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.

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 #109 is rated medium

What earns a board its rating is the shape of its opening. Here that opening offers four single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 34. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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 #109 despite the medium 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.

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 #109 is row 2.

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 #109?

Puzzle #109 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 #109 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #109, 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 #109?

Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.