9x9 Queens Puzzle #161

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 11-step reasoning is written out below.

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Step-by-step solution for puzzle #161

What follows is a complete solve of puzzle #161, worked the way a person works it rather than the way a computer does. It runs to 11 deductions — 9 queen placements and 2 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 confined to one line.

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

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

  1. Step 1. Only row 6, column 9 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 9, the rest of the red region and every square adjacent to it — diagonals included.

  2. Step 2. The orange region is down to one open square, at row 9, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 4, the remaining orange squares, and the ring of neighbours around row 9, column 4.

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

    Puzzle #161 after step 3: 3 of 9 queens placed
    The board after step 3 — 3 of 9 queens placed.

  4. Step 4. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.

  5. Step 5. 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: 1 square in all.

  6. Step 6. 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 1; khaki has one square left, so its queen takes row 8, column 2; pale yellow has one square left, so its queen takes row 7, column 5; amber has one square left, so its queen takes row 2, column 6; indigo has one square left, so its queen takes row 1, column 8; slate has one square left, so its queen takes row 5, 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 5, column 7.

    Puzzle #161 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 #161

Puzzle #161 is one of the hard-rated boards in the 9x9 set. Nine colors divide its 81 squares, and you win by placing nine queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.

Like every board on this site, #161 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 #161 are: red (1 square), orange (1 square), blue (2 squares), indigo (3 squares), mint (8 squares), amber (12 squares), khaki (13 squares), slate (16 squares), pale yellow (25 squares). An even split would hand every color 9 squares, so the red region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 25 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 red, with 1 square, is under far more pressure than pale yellow with 25. A 24-square gap like this one rewards players who start where the choices are fewest.

Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×3 frame; pale yellow is the loosest, occupying only 52% of its 6×8 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

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.

Red occupies 1 square within rows 6–6, columns 9–9. It never leaves row 6, so its queen is locked to that row.

Orange: 1 square across rows 9–9, columns 4–4. It never leaves row 9, so its queen is locked to that row.

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

Indigo occupies 3 squares within rows 1–1, columns 7–9. It never leaves row 1, so its queen is locked to that row.

Mint: 8 squares across rows 1–4, columns 1–2.

Amber covers 12 squares, spanning 3 rows and 6 columns (rows 1–3, columns 3–8).

Khaki occupies 13 squares within rows 5–9, columns 1–4.

Slate: 16 squares across rows 2–7, columns 5–9.

Pale yellow covers 25 squares, spanning 6 rows and 8 columns (rows 4–9, columns 2–9).

Where to start on this board

The most constrained color is where a solve should begin, and on #161 that is red with 1 square. It reaches from row 6 to row 6 and from column 9 to column 9 — 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 red never leaves row 6, its queen must be in that row — and that instantly settles row 6 for everyone else. Sweep along it and X out every square that is not red. 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. Indigo, blue and orange 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 #161

A quick way to find the next move is to count how many different colors touch each line. On this board, row 8 is the narrowest, touched by only 2 colors. Column 1 is the tightest column, with 2 colors.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 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 4 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

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 nine colors looking for one with a single square left; sweep the nine rows and nine columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 9x9 solve from a stalled one.

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 #161 is rated hard

The hard label comes out of four measurable things about #161: six of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 24. 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.

If #161 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 #161, start hunting for it near row 8.

As a sanity check rather than a spoiler: the finished solution to #161 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 #161?

Puzzle #161 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and six of its regions are confined to a single row or column, which gives you a forced move early.

Does 9x9 puzzle #161 have only one solution?

Yes. Every puzzle in the 9x9 collection, including #161, 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 #161?

Start with the red region — the most constrained color on this board, with 1 square spanning rows 6–6 and columns 9–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.