12x12 Queens Puzzle #108

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 15-step reasoning is written out below.

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Step-by-step solution for puzzle #108

What follows is a complete solve of puzzle #108, worked the way a person works it rather than the way a computer does. It runs to 15 deductions — 12 queen placements and 3 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: region shadow, pairing argument on rows, last square in a region.

Show the full walkthrough (4 steps) — contains the solution

  1. Step 1. Wherever amber ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.

  2. Step 2. Rose is down to 3 candidates in rows 11 and 12, columns 8 and 9, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about rose itself.

    Puzzle #108 after step 2: 0 of 12 queens placed
    The board after step 2 — 0 of 12 queens placed.

  3. Step 3. Teal and rose are now boxed into rows 11 and 12 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 15 squares at once.

  4. Step 4. From here the board cascades. Keep alternating the region sweep and the line sweep and 12 queens fall out in order without any new idea being needed: sea green has one square left, so its queen takes row 10, column 12; amber has one square left, so its queen takes row 2, column 11; blue has one square left, so its queen takes row 3, column 9; rose has one square left, so its queen takes row 12, column 8; orange has one square left, so its queen takes row 9, column 10; teal has one square left, so its queen takes row 11, column 2; pale yellow has one square left, so its queen takes row 7, column 3; indigo has one square left, so its queen takes row 1, column 4; khaki has one square left, so its queen takes row 8, column 5; mint has one square left, so its queen takes row 4, column 6; red has one square left, so its queen takes row 6, column 7; slate has one square left, so its queen takes row 5, column 1. 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 1.

    Puzzle #108 after step 4: 12 of 12 queens placed
    The board after step 4 — 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 #108

This is puzzle number 108 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 #108 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 #108 are: amber (3 squares), rose (3 squares), teal (4 squares), sea green (5 squares), slate (10 squares), blue (11 squares), pale yellow (12 squares), orange (12 squares), indigo (13 squares), red (23 squares), khaki (23 squares), mint (25 squares). An even split would hand every color 12 squares, so the amber region is running 9 squares under that average while the mint region is the roomiest area on the grid with 25 squares.

Every color takes one queen and one only, whatever its size — which is why the 22 square difference between amber and mint 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 sea green region is the most compact color here, filling 83% of the 3×2 box that encloses it, while slate is the most scattered at 42% of its 8×3 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

Below is every color on #108 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.

Amber: 3 squares across rows 1–2, columns 11–12.

Rose covers 3 squares, spanning 2 rows and 2 columns (rows 11–12, columns 8–9).

Teal occupies 4 squares within rows 11–12, columns 1–3.

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

Slate covers 10 squares, spanning 8 rows and 3 columns (rows 1–8, columns 1–3).

Blue occupies 11 squares within rows 3–8, columns 9–12.

Pale yellow: 12 squares across rows 6–11, columns 1–3.

Orange covers 12 squares, spanning 5 rows and 5 columns (rows 8–12, columns 8–12).

Indigo occupies 13 squares within rows 1–4, columns 2–5.

Red: 23 squares across rows 4–9, columns 5–11.

Khaki covers 23 squares, spanning 9 rows and 6 columns (rows 4–12, columns 3–8).

Mint occupies 25 squares within rows 1–6, columns 5–11.

Where to start on this board

The most constrained color is where a solve should begin, and on #108 that is amber with 3 squares. It reaches from row 1 to row 2 and from column 11 to column 12 — 2 rows by 2 columns. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.

Amber does not collapse onto a single line, so it will not hand you a queen on move one. What it does give you is a short list of 3 candidates. Take each one in turn, imagine the queen there, cross out its row, its column and the eight squares around it, and ask whether any other region is left with nowhere to go. One or two candidates usually fail that test immediately.

Work outward to the other flat regions next. Teal 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 #108

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3, 9 and 10 are the narrowest, touched by only 4 colors. Column 4 is the tightest column, with 2 colors.

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 4 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 11, meeting 6 colors, is the opposite case — the busiest line on #108 and the last one you should expect to crack.

Working through the middle game

Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.

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 #108 is rated medium

What earns a board its rating is the shape of its opening. Here that opening offers no single-line region at all, a 3-square smallest color, a tightest row touched by 4 colors, and a region-size spread of 22. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

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 #108 is column 4.

Here is a shape check that gives nothing away: when #108 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 #108?

Puzzle #108 is rated medium. Its smallest color region holds 3 squares, its tightest row is touched by 4 colors, and no region is confined to a single row or column, so the opening move takes some hunting.

Does 12x12 puzzle #108 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #108, 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 #108?

Start with the amber region — the most constrained color on this board, with 3 squares spanning rows 1–2 and columns 11–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.