10x10 Queens Puzzle #64

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 14-step reasoning is written out below.

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Step-by-step solution for puzzle #64

What follows is a complete solve of puzzle #64, worked the way a person works it rather than the way a computer does. It runs to 14 deductions — 10 queen placements and 4 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: last square in a region, region confined to one line, region shadow, pairing argument on rows, line confined to one region, last square in a column.

Show the full walkthrough (8 steps) — contains the solution

  1. Step 1. Only row 4, column 2 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 2, the rest of the mint region and every square adjacent to it — diagonals included.

  2. Step 2. The sea green region is down to one open square, at row 10, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 9, the remaining sea green squares, and the ring of neighbours around row 10, column 9.

  3. Step 3. Only row 8, column 10 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 10, in khaki and in the ring around that square is now dead.

  4. Step 4. All of the indigo 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: 3 squares come off.

    Puzzle #64 after step 4: 3 of 10 queens placed
    The board after step 4 — 3 of 10 queens placed.

  5. Step 5. Slate is down to 4 candidates in rows 3 and 5, columns 6–8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.

  6. Step 6. Here is the pairing argument. The indigo and amber regions have no open squares outside rows 1 and 2 — 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: 7 squares come off in one stroke.

  7. Step 7. Column 3 has only red squares left (3 open squares). The column needs a queen and only red can provide one, which pins that colour to this column and kills its squares elsewhere: 6 squares.

  8. Step 8. 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: column 4 is down to one square, putting a queen on row 3, column 4; slate has one square left, so its queen takes row 5, column 7; pale yellow has one square left, so its queen takes row 7, column 8; 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 1; orange has one square left, so its queen takes row 9, column 5; red has one square left, so its queen takes row 6, column 3. 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 6, column 3.

    Puzzle #64 after step 8: 10 of 10 queens placed
    The board after step 8 — 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 #64

This is puzzle number 64 in the 10x10 collection, rated hard. The grid holds 100 squares split between ten colored regions, and a finished board carries ten queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

Puzzle #64 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 #64 are: mint (1 square), sea green (1 square), indigo (5 squares), amber (7 squares), khaki (7 squares), pale yellow (9 squares), orange (12 squares), blue (15 squares), slate (17 squares), red (26 squares). An even split would hand every color 10 squares, so the mint region is running 9 squares under that average while the red region is the roomiest area on the grid with 26 squares.

Every color takes one queen and one only, whatever its size — which is why the 25 square difference between mint and red is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.

A region's outline can constrain it more than its size does. Here mint is the most solid, filling 100% of its 1×1 box, and pale yellow the most diffuse at 56% of its 4×4 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

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.

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

Sea green occupies 1 square within rows 10–10, columns 9–9. It never leaves row 10, so its queen is locked to that row.

Indigo: 5 squares across rows 1–4, columns 1–2.

Amber covers 7 squares, spanning 2 rows and 5 columns (rows 1–2, columns 6–10).

Khaki occupies 7 squares within rows 7–10, columns 8–10.

Pale yellow: 9 squares across rows 5–8, columns 6–9.

Orange covers 12 squares, spanning 4 rows and 5 columns (rows 7–10, columns 4–8).

Blue occupies 15 squares within rows 1–5, columns 2–6.

Slate: 17 squares across rows 2–7, columns 6–10.

Red covers 26 squares, spanning 8 rows and 5 columns (rows 3–10, columns 1–5).

Where to start on this board

Mint is the natural opening on #64: it is the smallest region on the grid at 1 square, boxed into rows 4–4 and columns 2–2. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.

Because mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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 and amber 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 #64

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 6 and 9 are the narrowest, touched by only 3 colors. Columns 1 and 3 are the tightest columns, with 2 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 7 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

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.

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 #64 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 #64 is rated hard

The hard label comes out of four measurable things about #64: four 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 25. 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.

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.

There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #64 is column 1.

Use this as a rough check on your progress: the completed board has 3 queens around the border and 7 queens in the middle, including 1 in a corner. 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 #64?

Puzzle #64 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #64 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #64, 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 #64?

Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.