10x10 Queens Puzzle #258

Difficulty: medium. 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 #258

What follows is a complete solve of puzzle #258, 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.

Show the full walkthrough (10 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 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 11 squares come off.

  3. Step 3. Pale yellow cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 8 squares in all.

    Puzzle #258 after step 3: 1 of 10 queens placed
    The board after step 3 — 1 of 10 queens placed.

  4. Step 4. The orange region is down to one open square, at row 9, column 1. Every colour holds exactly one queen, so that square is forced. Row 9 and column 1 are now spoken for, along with the squares touching that queen.

  5. Step 5. Only row 7, column 2 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 2, the rest of the pale yellow region and every square adjacent to it — diagonals included.

  6. Step 6. The slate region has 5 squares left, spread across rows 2, 3, 5, and 8, columns 4 and 5. 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.

  7. Step 7. Indigo, amber, and blue are now boxed into rows 1–3 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 4 squares at once.

    Puzzle #258 after step 7: 3 of 10 queens placed
    The board after step 7 — 3 of 10 queens placed.

  8. Step 8. All of the slate region's remaining squares (2 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.

  9. Step 9. Blue and khaki are now boxed into columns 9 and 10 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 5 squares at once.

  10. Step 10. 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: mint has one square left, so its queen takes row 4, column 8; amber has one square left, so its queen takes row 2, column 7; indigo has one square left, so its queen takes row 1, column 5; blue has one square left, so its queen takes row 3, column 10; khaki has one square left, so its queen takes row 8, column 9; slate has one square left, so its queen takes row 5, column 4; red has one square left, so its queen takes row 6, column 6. 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 6.

    Puzzle #258 after step 10: 10 of 10 queens placed
    The board after step 10 — 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 #258

Board #258 of the 10x10 bank carries a medium rating. Its 100 squares are shared between ten colored regions, and the solution places ten queens: one per row, one per column, one per region, none of them side by side or corner to corner.

Like every board on this site, #258 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 #258 are: sea green (1 square), amber (2 squares), pale yellow (2 squares), mint (6 squares), orange (6 squares), blue (7 squares), khaki (7 squares), indigo (11 squares), slate (22 squares), red (36 squares). An even split would hand every color 10 squares, so the sea green region is running 9 squares under that average while the red region is the roomiest area on the grid with 36 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; red has 36. The spread between them on this board is 35 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Shape matters as much as size. The amber region is the most compact color here, filling 100% of the 2×1 box that encloses it, while slate is the most scattered at 55% of its 8×5 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.

Region by region

Below is every color on #258 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: 1 square across rows 10–10, columns 3–3. It never leaves row 10, so its queen is locked to that row.

Amber covers 2 squares, spanning 2 rows and 1 column (rows 2–3, columns 7–7). It sits wholly inside column 7, so its queen must be in that column.

Pale yellow occupies 2 squares within rows 7–8, columns 2–2. It sits wholly inside column 2, so its queen must be in that column.

Mint: 6 squares across rows 3–5, columns 8–10.

Orange covers 6 squares, spanning 4 rows and 2 columns (rows 7–10, columns 1–2).

Blue occupies 7 squares within rows 1–3, columns 8–10.

Khaki: 7 squares across rows 6–10, columns 9–10.

Indigo covers 11 squares, spanning 2 rows and 8 columns (rows 1–2, columns 1–8).

Slate occupies 22 squares within rows 2–9, columns 1–5.

Red: 36 squares across rows 2–10, columns 4–9.

Where to start on this board

The most constrained color is where a solve should begin, and on #258 that is sea green with 1 square. It reaches from row 10 to row 10 and from column 3 to column 3 — 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 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 pale yellow 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 #258

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. Column 6 is the tightest column, with 2 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 8 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.

Keep alternating between two views of the board. In the region view you ask which of the ten colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 10x10 board unwinds.

The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.

Why puzzle #258 is rated medium

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 35. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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 #258 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 #258 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 #258?

Puzzle #258 is rated medium. 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 #258 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #258, 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 #258?

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.