10x10 Queens Puzzle #171

Difficulty: hard. Place 10 queens on this 10x10 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 16-step reasoning is written out below.

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Step-by-step solution for puzzle #171

What follows is a complete solve of puzzle #171, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 10 queen placements and 6 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 columns.

Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, pairing argument on columns, last square in a row.

Show the full walkthrough (9 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. All of the sea green region's remaining squares (3 open squares) lie in row 10. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.

  3. Step 3. Only row 8, column 1 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 1, in khaki and in the ring around that square is now dead.

    Puzzle #171 after step 3: 2 of 10 queens placed
    The board after step 3 — 2 of 10 queens placed.

  4. Step 4. All of the blue region's remaining squares (3 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.

  5. Step 5. Red is down to 3 candidates in rows 6 and 7, columns 3 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.

  6. Step 6. Here is the pairing argument. The red and pale yellow regions have no open squares outside rows 6 and 7 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 6 squares come off in one stroke.

    Puzzle #171 after step 6: 2 of 10 queens placed
    The board after step 6 — 2 of 10 queens placed.

  7. Step 7. Wherever slate ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.

  8. Step 8. Here is the pairing argument. The mint and red regions have no open squares outside columns 3 and 4 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 5 squares come off in one stroke.

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

    Puzzle #171 after step 9: 10 of 10 queens placed
    The board after step 9 — 10 of 10 queens placed.

That completes the board: 10 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 10x10 Queens puzzle #171

10x10 puzzle #171 is rated hard. Ten colored regions cover its 100 squares, and your job is to seat ten queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

Puzzle #171 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 #171 are: orange (1 square), khaki (5 squares), blue (6 squares), sea green (7 squares), amber (9 squares), mint (10 squares), slate (11 squares), red (14 squares), pale yellow (16 squares), indigo (21 squares). An even split would hand every color 10 squares, so the orange region is running 9 squares under that average while the indigo region is the roomiest area on the grid with 21 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Orange has only 1 place to put its queen; indigo has 21. The spread between them on this board is 20 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Square count is only half the story — how a color is arranged counts too. Blue is the tidiest shape on #171, packing 100% of its 3×2 bounding box, whereas mint straggles across its 4×5 box at just 50%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

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.

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

Khaki covers 5 squares, spanning 3 rows and 2 columns (rows 8–10, columns 1–2).

Blue occupies 6 squares within rows 1–3, columns 1–2.

Sea green: 7 squares across rows 9–10, columns 3–6.

Amber covers 9 squares, spanning 4 rows and 4 columns (rows 1–4, columns 3–6).

Mint occupies 10 squares within rows 3–6, columns 1–5.

Slate: 11 squares across rows 3–8, columns 8–10.

Red covers 14 squares, spanning 5 rows and 5 columns (rows 5–9, columns 1–5).

Pale yellow occupies 16 squares within rows 6–10, columns 5–10.

Indigo: 21 squares across rows 1–5, columns 5–10.

Where to start on this board

Open on the tightest color. Here that is orange, confined to 1 square between rows 9 and 9 and columns 7 and 7. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #171.

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. Sea green 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 #171

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 7 and 10 are the narrowest, touched by only 3 colors. Columns 7, 8, 9 and 10 are the tightest columns, with 3 colors.

Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 9 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.

Working through the middle game

The middle of #171 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.

With ten queens to seat and ten regions competing over 100 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 10x10 boards quickly and players who stall halfway.

Resist the plausible-looking queen. On #171 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 #171 is rated hard

The hard label comes out of four measurable things about #171: two of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 20. Boards that score tight on those measures give up their first queen fast.

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.

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 10x10

At 10x10 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.

One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #171 the place to look for one is around row 1.

Use this as a rough check on your progress: the completed board has 4 queens around the border and 6 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.

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 10x10 Queens puzzle #171?

Puzzle #171 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #171 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #171, has been verified to have exactly one valid arrangement of ten queens, and it can always be reached by logic alone — no guessing or backtracking needed.

Where should I place my first queen on puzzle #171?

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.