10x10 Queens Puzzle #286

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 #286

What follows is a complete solve of puzzle #286, 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 region confined to one line.

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

Show the full walkthrough (7 steps) — contains the solution

  1. Step 1. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 10 squares in all.

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

  3. Step 3. Only row 3, column 7 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 7, in blue 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 row 4. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.

    Puzzle #286 after step 4: 2 of 10 queens placed
    The board after step 4 — 2 of 10 queens placed.

  5. Step 5. Slate cannot leave row 5 — every one of its open squares (2 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 4 squares in all.

  6. Step 6. All of the khaki region's remaining squares (3 open squares) lie in column 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 3 squares come off.

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

    Puzzle #286 after step 7: 10 of 10 queens placed
    The board after step 7 — 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 #286

Board #286 of the 10x10 bank carries a hard 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, #286 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 #286 are: indigo (2 squares), khaki (5 squares), amber (6 squares), blue (6 squares), orange (6 squares), slate (9 squares), mint (10 squares), pale yellow (14 squares), red (18 squares), sea green (24 squares). An even split would hand every color 10 squares, so the indigo region is running 8 squares under that average while the sea green region is the roomiest area on the grid with 24 squares.

Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So indigo, with 2 squares, is under far more pressure than sea green with 24. A 22-square gap like this one rewards players who start where the choices are fewest.

Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×2 box that encloses it, while red is the most scattered at 38% of its 6×8 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 #286 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.

Indigo covers 2 squares, spanning 1 row and 2 columns (rows 1–1, columns 4–5). It never leaves row 1, so its queen is locked to that row.

Khaki occupies 5 squares within rows 6–8, columns 9–10.

Amber: 6 squares across rows 1–2, columns 6–10.

Blue covers 6 squares, spanning 2 rows and 4 columns (rows 2–3, columns 5–8).

Orange occupies 6 squares within rows 7–10, columns 1–2.

Slate: 9 squares across rows 2–5, columns 8–10.

Mint covers 10 squares, spanning 4 rows and 4 columns (rows 1–4, columns 1–4).

Pale yellow occupies 14 squares within rows 2–7, columns 1–5.

Red: 18 squares across rows 3–8, columns 2–9.

Sea green covers 24 squares, spanning 5 rows and 9 columns (rows 6–10, columns 2–10).

Where to start on this board

The most constrained color is where a solve should begin, and on #286 that is indigo with 2 squares. It reaches from row 1 to row 1 and from column 4 to column 5 — 1 row by 2 columns. 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 indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Amber and blue 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 #286

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

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 #286 is rated hard

What earns a board its rating is the shape of its opening. Here that opening offers one single-line region, a 2-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 22. 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.

If #286 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.

Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #286, start hunting for it near row 9.

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 #286?

Puzzle #286 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 2 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #286 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #286, 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 #286?

Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–1 and columns 4–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.