12x12 Queens Puzzle #196

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 22-step reasoning is written out below.

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Step-by-step solution for puzzle #196

What follows is a complete solve of puzzle #196, worked the way a person works it rather than the way a computer does. It runs to 22 deductions — 12 queen placements and 10 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: last square in a region, region shadow, region confined to one line, pairing argument on rows, last square in a row.

Show the full walkthrough (15 steps) — contains the solution

  1. Step 1. Only row 1, column 10 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 10, the rest of the indigo region and every square adjacent to it — diagonals included.

  2. Step 2. Blue is down to 3 candidates in rows 2 and 3, columns 11 and 12, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.

  3. Step 3. Rose cannot leave column 2 — every one of its open squares (3 open squares) sits there. That column belongs to rose, so every square in it owned by another colour is dead: 10 squares in all.

  4. Step 4. Here is the pairing argument. The amber, blue, and mint regions have no open squares outside rows 2–4 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 12 squares come off in one stroke.

  5. Step 5. Slate is down to 5 candidates in rows 5–7, columns 3–5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.

    Puzzle #196 after step 5: 1 of 12 queens placed
    The board after step 5 — 1 of 12 queens placed.

  6. Step 6. All of the pale yellow region's remaining squares (3 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.

  7. Step 7. Slate, red, and pale yellow are now boxed into rows 5–7 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 17 squares at once.

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

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

  10. Step 10. Here is the pairing argument. The khaki and orange regions have no open squares outside rows 8 and 9 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 9 squares come off in one stroke.

    Puzzle #196 after step 10: 1 of 12 queens placed
    The board after step 10 — 1 of 12 queens placed.

  11. Step 11. Only row 10, column 3 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 3, in sea green and in the ring around that square is now dead.

  12. Step 12. The rose region is down to one open square, at row 12, column 2. Every colour holds exactly one queen, so that square is forced. Row 12 and column 2 are now spoken for, along with the squares touching that queen.

  13. Step 13. Look along row 7: one square survives, row 7, column 1, which belongs to pale yellow. A row cannot be left empty, so the queen goes there. Everything else in row 7, in column 1, in pale yellow and in the ring around that square is now dead.

  14. Step 14. All of the amber region's remaining squares (3 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 3 squares come off.

  15. Step 15. 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: slate has one square left, so its queen takes row 5, column 5; red has one square left, so its queen takes row 6, column 12; blue has one square left, so its queen takes row 3, column 11; amber has one square left, so its queen takes row 2, column 4; mint has one square left, so its queen takes row 4, column 9; orange has one square left, so its queen takes row 9, column 8; khaki has one square left, so its queen takes row 8, column 6; teal has one square left, so its queen takes row 11, column 7. 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 11, column 7.

    Puzzle #196 after step 15: 12 of 12 queens placed
    The board after step 15 — 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 #196

This is puzzle number 196 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.

Puzzle #196 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 #196 are: indigo (1 square), rose (3 squares), blue (8 squares), red (8 squares), amber (11 squares), slate (11 squares), sea green (11 squares), mint (13 squares), khaki (13 squares), orange (13 squares), pale yellow (14 squares), teal (38 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the teal region is the roomiest area on the grid with 38 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 1 square while teal has 38 to spare, a gap of 37. 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 1×1 box that encloses it, while slate is the most scattered at 37% of its 6×5 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

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.

Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 10–10). It never leaves row 1, so its queen is locked to that row.

Rose occupies 3 squares within rows 10–12, columns 2–2. It sits wholly inside column 2, so its queen must be in that column.

Blue: 8 squares across rows 1–3, columns 10–12.

Red covers 8 squares, spanning 3 rows and 4 columns (rows 4–6, columns 9–12).

Amber occupies 11 squares within rows 1–4, columns 1–5.

Slate: 11 squares across rows 2–7, columns 3–7.

Sea green covers 11 squares, spanning 5 rows and 4 columns (rows 8–12, columns 1–4).

Mint occupies 13 squares within rows 1–4, columns 6–10.

Khaki: 13 squares across rows 5–9, columns 4–7.

Orange covers 13 squares, spanning 6 rows and 4 columns (rows 4–9, columns 7–10).

Pale yellow occupies 14 squares within rows 2–8, columns 1–3.

Teal: 38 squares across rows 6–12, columns 3–12.

Where to start on this board

The most constrained color is where a solve should begin, and on #196 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 10 to column 10 — 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 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.

Nothing on this grid is pinned to one or two lines, which is why the opening takes patience. Scan the columns for one where only two or three colors appear — whichever region is forced into it claims that column outright. Chaining two of those observations is normally enough to break #196 open.

The tightest lines on puzzle #196

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 10, 11 and 12 are the narrowest, touched by only 3 colors. Columns 1, 8, 11 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 6 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.

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 #196 is rated hard

The hard label comes out of four measurable things about #196: three of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 37. 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.

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.

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 10 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 #196?

Puzzle #196 is rated hard. 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 #196 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #196, 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 #196?

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