10x10 Queens Puzzle #59

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 #59

What follows is a complete solve of puzzle #59, 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 region shadow.

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

Show the full walkthrough (12 steps) — contains the solution

  1. Step 1. Only row 2, column 3 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 3, the rest of the amber region and every square adjacent to it — diagonals included.

  2. Step 2. All of the blue 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: 7 squares come off.

  3. Step 3. Only row 10, column 2 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 2, in sea green and in the ring around that square is now dead.

  4. Step 4. All of the mint 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: 8 squares come off.

    Puzzle #59 after step 4: 2 of 10 queens placed
    The board after step 4 — 2 of 10 queens placed.

  5. Step 5. Indigo cannot leave row 1 — every one of its open squares (5 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 1 square in all.

  6. Step 6. The blue region is down to one open square, at row 3, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 1, the remaining blue squares, and the ring of neighbours around row 3, column 1.

  7. Step 7. Only row 4, column 10 is still open in mint, and since each colour must take a queen, that settles it. Everything else in row 4, in column 10, in mint and in the ring around that square is now dead.

  8. Step 8. The slate region is down to one open square, at row 5, column 4. Every colour holds exactly one queen, so that square is forced. Row 5 and column 4 are now spoken for, along with the squares touching that queen.

    Puzzle #59 after step 8: 5 of 10 queens placed
    The board after step 8 — 5 of 10 queens placed.

  9. Step 9. Red cannot leave column 9 — every one of its open squares (2 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 5 squares in all.

  10. Step 10. Wherever pale yellow 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.

  11. Step 11. Khaki is down to 5 candidates in rows 6–9, columns 7 and 8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.

  12. Step 12. 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: orange has one square left, so its queen takes row 9, column 6; pale yellow has one square left, so its queen takes row 7, column 5; red has one square left, so its queen takes row 6, column 9; khaki has one square left, so its queen takes row 8, column 8; indigo has one square left, so its queen takes row 1, 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 1, column 7.

    Puzzle #59 after step 12: 10 of 10 queens placed
    The board after step 12 — 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 #59

10x10 puzzle #59 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.

Like every board on this site, #59 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 #59 are: amber (1 square), mint (3 squares), sea green (4 squares), orange (6 squares), blue (7 squares), slate (8 squares), indigo (12 squares), khaki (15 squares), red (19 squares), pale yellow (25 squares). An even split would hand every color 10 squares, so the amber region is running 9 squares under that average while the pale yellow region is the roomiest area on the grid with 25 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Amber has only 1 place to put its queen; pale yellow has 25. The spread between them on this board is 24 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 #59, packing 100% of its 1×1 bounding box, whereas indigo straggles across its 4×7 box at just 43%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.

Amber occupies 1 square within rows 2–2, columns 3–3. It never leaves row 2, so its queen is locked to that row.

Mint: 3 squares across rows 2–4, columns 10–10. It sits wholly inside column 10, so its queen must be in that column.

Sea green covers 4 squares, spanning 2 rows and 3 columns (rows 9–10, columns 1–3).

Orange occupies 6 squares within rows 8–10, columns 4–7.

Blue: 7 squares across rows 1–3, columns 1–3.

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

Indigo occupies 12 squares within rows 1–4, columns 4–10.

Khaki: 15 squares across rows 6–10, columns 7–10.

Red covers 19 squares, spanning 6 rows and 6 columns (rows 2–7, columns 5–10).

Pale yellow occupies 25 squares within rows 4–9, columns 1–6.

Where to start on this board

Open on the tightest color. Here that is amber, confined to 1 square between rows 2 and 2 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 #59.

Because amber never leaves row 2, its queen must be in that row — and that instantly settles row 2 for everyone else. Sweep along it and X out every square that is not amber. 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 #59

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, 8 and 9 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 2 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 2 sits at the far end with 6 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.

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 #59 is rated hard

Ratings track how generous the opening is. On #59 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 24-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.

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.

If #59 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 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 #59 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 #59?

Puzzle #59 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #59 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #59, 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 #59?

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