12x12 Queens Puzzle #112

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 19-step reasoning is written out below.

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Step-by-step solution for puzzle #112

What follows is a complete solve of puzzle #112, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 12 queen placements and 7 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 (10 steps) — contains the solution

  1. Step 1. Only row 12, column 5 is still open in rose, and since each colour must take a queen, that settles it. That clears row 12, column 5, the rest of the rose region and every square adjacent to it — diagonals included.

  2. Step 2. Amber is down to 4 candidates in rows 1–3, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.

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

    Puzzle #112 after step 3: 1 of 12 queens placed
    The board after step 3 — 1 of 12 queens placed.

  4. Step 4. Wherever mint ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

  5. Step 5. Sea green is down to 5 candidates in rows 10 and 11, columns 1–3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about sea green itself.

  6. Step 6. The teal region has 3 squares left, spread across rows 10 and 11, columns 11 and 12. 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.

  7. Step 7. Indigo and amber are now boxed into rows 1 and 2 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 6 squares at once.

    Puzzle #112 after step 7: 1 of 12 queens placed
    The board after step 7 — 1 of 12 queens placed.

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

  9. Step 9. All of the red 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: 3 squares come off.

  10. Step 10. 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: orange has one square left, so its queen takes row 9, column 8; red has one square left, so its queen takes row 6, column 10; khaki has one square left, so its queen takes row 8, column 6; indigo has one square left, so its queen takes row 1, column 7; pale yellow has one square left, so its queen takes row 7, 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 7, column 2.

    Puzzle #112 after step 10: 12 of 12 queens placed
    The board after step 10 — 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 #112

This is puzzle number 112 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.

There is exactly one way to finish #112, 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 #112 are: rose (1 square), blue (2 squares), amber (4 squares), mint (5 squares), teal (6 squares), sea green (11 squares), slate (14 squares), khaki (16 squares), orange (16 squares), indigo (19 squares), pale yellow (23 squares), red (27 squares). An even split would hand every color 12 squares, so the rose region is running 11 squares under that average while the red region is the roomiest area on the grid with 27 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Rose is working with 1 square while red has 27 to spare, a gap of 26. Attack the cramped end and the roomy regions tend to resolve themselves.

Shape matters as much as size. The blue region is the most compact color here, filling 100% of the 1×2 box that encloses it, while khaki is the most scattered at 46% of its 7×5 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

Here is the full footprint of every color on puzzle #112, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Rose covers 1 square, spanning 1 row and 1 column (rows 12–12, columns 5–5). It never leaves row 12, so its queen is locked to that row.

Blue occupies 2 squares within rows 3–3, columns 9–10. It never leaves row 3, so its queen is locked to that row.

Amber: 4 squares across rows 1–3, columns 1–2.

Mint covers 5 squares, spanning 3 rows and 2 columns (rows 2–4, columns 11–12).

Teal occupies 6 squares within rows 10–12, columns 10–12.

Sea green: 11 squares across rows 10–12, columns 1–5.

Slate covers 14 squares, spanning 5 rows and 6 columns (rows 2–6, columns 2–7).

Khaki occupies 16 squares within rows 6–12, columns 4–8.

Orange: 16 squares across rows 7–12, columns 7–10.

Indigo covers 19 squares, spanning 3 rows and 10 columns (rows 1–3, columns 3–12).

Pale yellow occupies 23 squares within rows 4–10, columns 1–7.

Red: 27 squares across rows 4–10, columns 5–12.

Where to start on this board

The most constrained color is where a solve should begin, and on #112 that is rose with 1 square. It reaches from row 12 to row 12 and from column 5 to column 5 — 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 rose never leaves row 12, its queen must be in that row — and that instantly settles row 12 for everyone else. Sweep along it and X out every square that is not rose. 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. Blue 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 #112

A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Column 1 is the tightest column, with 3 colors.

A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 11 rows. Row 10 is the crowded extreme at 6 colors — worth revisiting later, not now.

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.

Resist the plausible-looking queen. On #112 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.

Why puzzle #112 is rated medium

The rating reflects how much the board concedes up front. Puzzle #112 has three regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 26 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a medium board from the rest.

Stuck on #112 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.

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.

A non-spoiling distribution check: solved, #112 carries 4 queens on the perimeter against 8 queens inside, with the corners empty. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.

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 #112?

Puzzle #112 is rated medium. 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 #112 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #112, 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 #112?

Start with the rose region — the most constrained color on this board, with 1 square spanning rows 12–12 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.