9x9 Queens Puzzle #123

Difficulty: medium. 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 17-step reasoning is written out below.

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Step-by-step solution for puzzle #123

What follows is a complete solve of puzzle #123, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 9 queen placements and 8 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, region confined to one line, pairing argument on rows, last square in a region, last square in a row, last square in a column.

Show the full walkthrough (13 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. Blue is down to 4 candidates in rows 2 and 3, columns 5–7, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about blue itself.

  3. Step 3. Amber cannot leave column 9 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 6 squares in all.

  4. Step 4. Wherever red ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

    Puzzle #123 after step 4: 0 of 9 queens placed
    The board after step 4 — 0 of 9 queens placed.

  5. Step 5. Khaki cannot leave row 8 — every one of its open squares (3 open squares) sits there. That row belongs to khaki, so every square on it owned by another colour is dead: 7 squares in all.

  6. Step 6. Here is the pairing argument. The indigo, amber, and blue regions have no open squares outside rows 1–3 — 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: 11 squares come off in one stroke.

  7. Step 7. Only row 4, column 1 is still open in mint, and since each colour must take a queen, that settles it. Everything else in row 4, in column 1, in mint and in the ring around that square is now dead.

  8. Step 8. All of the pale yellow region's remaining squares (2 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 3 squares come off.

  9. Step 9. Only row 8, column 4 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 4, the rest of the khaki region and every square adjacent to it — diagonals included.

    Puzzle #123 after step 9: 2 of 9 queens placed
    The board after step 9 — 2 of 9 queens placed.

  10. Step 10. Row 3 has a single open square left, at column 7 in the blue region. Every row needs a queen, so this one is forced. Row 3 and column 7 are now spoken for, along with the squares touching that queen.

  11. Step 11. Look along row 2: one square survives, row 2, column 9, which belongs to amber. A row cannot be left empty, so the queen goes there. That clears row 2, column 9, the rest of the amber region and every square adjacent to it — diagonals included.

  12. Step 12. All of the orange region's remaining squares (2 open squares) lie in row 9. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 1 square come off.

  13. Step 13. 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: pale yellow has one square left, so its queen takes row 7, column 2; column 3 is down to one square, putting a queen on row 5, column 3; red has one square left, so its queen takes row 6, column 6; indigo has one square left, so its queen takes row 1, column 5; orange has one square left, so its queen takes row 9, column 8. 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 9, column 8.

    Puzzle #123 after step 13: 9 of 9 queens placed
    The board after step 13 — 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 #123

9x9 puzzle #123 is rated medium. Nine colored regions cover its 81 squares, and your job is to seat nine queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

There is exactly one way to finish #123, 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 nine regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #123 are: amber (3 squares), khaki (3 squares), blue (4 squares), red (5 squares), indigo (7 squares), pale yellow (8 squares), mint (10 squares), orange (13 squares), slate (28 squares). An even split would hand every color 9 squares, so the amber region is running 6 squares under that average while the slate region is the roomiest area on the grid with 28 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 3 squares while slate has 28 to spare, a gap of 25. Attack the cramped end and the roomy regions tend to resolve themselves.

Shape matters as much as size. The khaki region is the most compact color here, filling 100% of the 1×3 box that encloses it, while indigo is the most scattered at 47% of its 3×5 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 medium board.

Amber: 3 squares across rows 1–2, columns 8–9.

Khaki covers 3 squares, spanning 1 row and 3 columns (rows 8–8, columns 2–4). It never leaves row 8, so its queen is locked to that row.

Blue occupies 4 squares within rows 2–3, columns 5–7.

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

Indigo covers 7 squares, spanning 3 rows and 5 columns (rows 1–3, columns 4–8).

Pale yellow occupies 8 squares within rows 5–9, columns 1–3.

Mint: 10 squares across rows 1–4, columns 1–3.

Orange covers 13 squares, spanning 3 rows and 6 columns (rows 7–9, columns 4–9).

Slate occupies 28 squares within rows 3–8, columns 2–9.

Where to start on this board

Amber is the natural opening on #123: it is the smallest region on the grid at 3 squares, boxed into rows 1–2 and columns 8–9. Since one of those squares must hold a queen, that block of 2 rows and 2 columns is the most contested part of the board.

Amber spans 2 rows, so it stays open for now — but it is still the shortest list on the board at 3 squares. Work through those squares first and mark the ones that would leave another region homeless. That pruning is what makes the rest of the grid legible.

Work outward to the other flat regions next. Khaki and blue 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 #123

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

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 8 rows. Row 8, meeting 4 colors, is the opposite case — the busiest line on #123 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.

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.

Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.

Why puzzle #123 is rated medium

The medium label comes out of four measurable things about #123: one of its regions is locked to one row or column, its smallest color spans 3 squares, its least crowded row meets 2 colors, and its largest and smallest regions differ by 25. Boards that score tight on those measures give up their first queen fast.

Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.

Stuck on #123 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 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.

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 4 on this board.

Here is a shape check that gives nothing away: when #123 is solved, 4 queens sit on the outer edge of the grid and 5 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 9x9 Queens puzzle #123?

Puzzle #123 is rated medium. Its smallest color region holds 3 squares, its tightest row is touched by 2 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.

Does 9x9 puzzle #123 have only one solution?

Yes. Every puzzle in the 9x9 collection, including #123, 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 #123?

Start with the amber region — the most constrained color on this board, with 3 squares spanning rows 1–2 and columns 8–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.