9x9 Queens Puzzle #104

Difficulty: hard. Place 9 queens on this 9x9 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 12-step reasoning is written out below.

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Step-by-step solution for puzzle #104

What follows is a complete solve of puzzle #104, worked the way a person works it rather than the way a computer does. It runs to 12 deductions — 9 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 region shadow.

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

Show the full walkthrough (6 steps) — contains the solution

  1. Step 1. Only row 9, column 7 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 7, the rest of the orange region and every square adjacent to it — diagonals included.

  2. Step 2. The indigo region is down to one open square, at row 1, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 8, the remaining indigo squares, and the ring of neighbours around row 1, column 8.

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

    Puzzle #104 after step 3: 2 of 9 queens placed
    The board after step 3 — 2 of 9 queens placed.

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

  5. Step 5. Red cannot leave column 6 — every one of its open squares (2 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 5 squares in all.

  6. Step 6. 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: khaki has one square left, so its queen takes row 8, column 5; red has one square left, so its queen takes row 6, column 6; 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; amber has one square left, so its queen takes row 2, column 1; pale yellow has one square left, so its queen takes row 7, column 2; mint has one square left, so its queen takes row 4, 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 4, column 9.

    Puzzle #104 after step 6: 9 of 9 queens placed
    The board after step 6 — 9 of 9 queens placed.

That completes the board: 9 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 9x9 Queens puzzle #104

This is puzzle number 104 in the 9x9 collection, rated hard. The grid holds 81 squares split between nine colored regions, and a finished board carries nine queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

Like every board on this site, #104 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.

How the colors divide this board

From smallest to largest, the regions on puzzle #104 are: orange (1 square), red (2 squares), indigo (3 squares), amber (4 squares), khaki (8 squares), blue (11 squares), slate (13 squares), pale yellow (19 squares), mint (20 squares). An even split would hand every color 9 squares, so the orange region is running 8 squares under that average while the mint region is the roomiest area on the grid with 20 squares.

Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So orange, with 1 square, is under far more pressure than mint with 20. A 19-square gap like this one rewards players who start where the choices are fewest.

Square count is only half the story — how a color is arranged counts too. Amber is the tidiest shape on #104, packing 100% of its 2×2 bounding box, whereas blue straggles across its 5×5 box at just 44%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Here is the full footprint of every color on puzzle #104, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Orange occupies 1 square within rows 9–9, columns 7–7. It never leaves row 9, so its queen is locked to that row.

Red: 2 squares across rows 6–7, columns 6–6. It sits wholly inside column 6, so its queen must be in that column.

Indigo covers 3 squares, spanning 2 rows and 2 columns (rows 1–2, columns 7–8).

Amber occupies 4 squares within rows 1–2, columns 1–2.

Khaki: 8 squares across rows 6–9, columns 5–7.

Blue covers 11 squares, spanning 5 rows and 5 columns (rows 1–5, columns 1–5).

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

Pale yellow: 19 squares across rows 4–9, columns 1–4.

Mint covers 20 squares, spanning 9 rows and 3 columns (rows 1–9, columns 7–9).

Where to start on this board

Begin where the choices are fewest. On this board that means orange, which owns just 1 square inside a 1×1 window running from row 9 to row 9 and column 7 to column 7. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

Because orange never leaves row 9, its queen must be in that row — and that instantly settles row 9 for everyone else. Sweep along it and X out every square that is not orange. 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. Amber, indigo and red 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 #104

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 8 are the narrowest, touched by only 3 colors. Column 9 is the tightest column, with 1 color.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 8 rows. By contrast row 6 meets 5 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

Working through the middle game

Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.

Two checks, run alternately, will carry you through the 81 squares of this grid. First: has any of the nine regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 9x9 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 #104 is rated hard

The rating reflects how much the board concedes up front. Puzzle #104 has three regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 19 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

If #104 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.

Techniques that work well at 9x9

At 9x9 a full re-scan after every move starts to get slow, so work locally instead. Once a queen goes down, check only the regions, rows and columns that placement actually touched — that is where new information appeared. Fall back to a complete sweep only when the local pass turns up nothing, and your solve times will drop noticeably.

Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #104, start hunting for it near column 9.

As a sanity check rather than a spoiler: the finished solution to #104 puts 4 queens on the outer border and 5 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.

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 9x9 Queens puzzle #104?

Puzzle #104 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 9x9 puzzle #104 have only one solution?

Yes. Every puzzle in the 9x9 collection, including #104, has been verified to have exactly one valid arrangement of nine queens, and it can always be reached by logic alone — no guessing or backtracking needed.

Where should I place my first queen on puzzle #104?

Start with the orange region — the most constrained color on this board, with 1 square spanning rows 9–9 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.