10x10 Queens Puzzle #71

Difficulty: easy. 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 19-step reasoning is written out below.

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Step-by-step solution for puzzle #71

What follows is a complete solve of puzzle #71, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 10 queen placements and 9 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, line confined to one region, pairing argument on columns, last square in a row.

Show the full walkthrough (11 steps) — contains the solution

  1. Step 1. Only row 10, column 3 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 3, the rest of the sea green region and every square adjacent to it — diagonals included.

  2. Step 2. All of the amber region's remaining squares (2 open squares) lie in column 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 9 squares come off.

  3. Step 3. The indigo region has 3 squares left, spread across rows 1 and 2, columns 8 and 9. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.

  4. Step 4. All of the red region's remaining squares (2 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 9 squares come off.

    Puzzle #71 after step 4: 1 of 10 queens placed
    The board after step 4 — 1 of 10 queens placed.

  5. Step 5. Khaki is down to 2 candidates in rows 8 and 9, columns 2 and 5, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about khaki itself.

  6. Step 6. Here is the pairing argument. The khaki and orange regions have no open squares outside rows 8 and 9 — 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: 3 squares come off in one stroke.

  7. Step 7. Red and pale yellow are now boxed into rows 6 and 7 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 8 squares at once.

    Puzzle #71 after step 7: 1 of 10 queens placed
    The board after step 7 — 1 of 10 queens placed.

  8. Step 8. Slate is down to 6 candidates in rows 4 and 5, columns 2, 5, 6, 7, and 8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.

  9. Step 9. Column 4 has only blue squares left (5 open squares). The column needs a queen and only blue can provide one, which pins that colour to this column and kills its squares elsewhere: 11 squares.

  10. Step 10. Here is the pairing argument. The indigo, mint, and pale yellow regions have no open squares outside columns 7–9 — 3 colours confined to 3 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.

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

    Puzzle #71 after step 11: 10 of 10 queens placed
    The board after step 11 — 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 #71

10x10 puzzle #71 is rated easy. 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 #71 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 #71 are: sea green (1 square), amber (2 squares), red (2 squares), indigo (5 squares), orange (8 squares), pale yellow (11 squares), mint (13 squares), khaki (13 squares), slate (19 squares), blue (26 squares). An even split would hand every color 10 squares, so the sea green region is running 9 squares under that average while the blue region is the roomiest area on the grid with 26 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Sea green has only 1 place to put its queen; blue has 26. The spread between them on this board is 25 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. Amber is the tidiest shape on #71, packing 100% of its 2×1 bounding box, whereas mint straggles across its 6×4 box at just 54%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Below is every color on #71 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.

Sea green occupies 1 square within rows 10–10, columns 3–3. It never leaves row 10, so its queen is locked to that row.

Amber: 2 squares across rows 2–3, columns 10–10. It sits wholly inside column 10, so its queen must be in that column.

Red covers 2 squares, spanning 2 rows and 1 column (rows 6–7, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Indigo occupies 5 squares within rows 1–2, columns 8–10.

Orange: 8 squares across rows 8–10, columns 6–9.

Pale yellow covers 11 squares, spanning 5 rows and 4 columns (rows 6–10, columns 7–10).

Mint occupies 13 squares within rows 2–7, columns 7–10.

Khaki: 13 squares across rows 7–10, columns 1–5.

Slate covers 19 squares, spanning 5 rows and 7 columns (rows 4–8, columns 2–8).

Blue occupies 26 squares within rows 1–5, columns 1–7.

Where to start on this board

Open on the tightest color. Here that is sea green, confined to 1 square between rows 10 and 10 and columns 3 and 3. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #71.

Because sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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, amber 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 #71

A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Columns 1, 2, 4, 5 and 6 are the tightest columns, with 3 colors.

A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 9 rows. Row 7 is the crowded extreme at 5 colors — worth revisiting later, not now.

Working through the middle game

The middle of #71 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.

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 #71 is rated easy

What earns a board its rating is the shape of its opening. Here that opening offers four single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 25. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

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 #71 the place to look for one is around row 1.

A non-spoiling distribution check: solved, #71 carries 4 queens on the perimeter against 6 queens inside, with the corners empty. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.

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 #71?

Puzzle #71 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #71 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #71, 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 #71?

Start with the sea green region — the most constrained color on this board, with 1 square spanning rows 10–10 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.