10x10 Queens Puzzle #259

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 15-step reasoning is written out below.

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Step-by-step solution for puzzle #259

What follows is a complete solve of puzzle #259, worked the way a person works it rather than the way a computer does. It runs to 15 deductions — 10 queen placements and 5 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: region shadow, last square in a region, region confined to one line, pairing argument on rows.

Show the full walkthrough (10 steps) — contains the solution

  1. Step 1. Wherever indigo ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

  2. Step 2. The blue region is down to one open square, at row 3, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 2, the remaining blue squares, and the ring of neighbours around row 3, column 2.

  3. Step 3. Only row 5, column 1 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 1, in slate and in the ring around that square is now dead.

    Puzzle #259 after step 3: 2 of 10 queens placed
    The board after step 3 — 2 of 10 queens placed.

  4. Step 4. The indigo region is down to one open square, at row 1, column 3. Every colour holds exactly one queen, so that square is forced. Row 1 and column 3 are now spoken for, along with the squares touching that queen.

  5. Step 5. Amber cannot leave row 2 — every one of its open squares (2 open squares) sits there. That row belongs to amber, 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 (2 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 8 squares come off.

  7. Step 7. Wherever sea green ends up among its 3 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.

    Puzzle #259 after step 7: 3 of 10 queens placed
    The board after step 7 — 3 of 10 queens placed.

  8. Step 8. The khaki region is down to one open square, at row 8, column 9. Every colour holds exactly one queen, so that square is forced. Row 8 and column 9 are now spoken for, along with the squares touching that queen.

  9. Step 9. Mint and red are now boxed into rows 4 and 6 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 6 squares at once.

  10. Step 10. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: pale yellow has one square left, so its queen takes row 7, column 7; amber has one square left, so its queen takes row 2, column 5; mint has one square left, so its queen takes row 4, column 4; red has one square left, so its queen takes row 6, column 10; orange has one square left, so its queen takes row 9, column 6; sea green has one square left, so its queen takes row 10, column 8. 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 8.

    Puzzle #259 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 #259

10x10 puzzle #259 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.

Puzzle #259 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 #259 are: blue (2 squares), khaki (2 squares), slate (3 squares), indigo (4 squares), sea green (7 squares), amber (8 squares), pale yellow (11 squares), mint (18 squares), red (19 squares), orange (26 squares). An even split would hand every color 10 squares, so the blue region is running 8 squares under that average while the orange region is the roomiest area on the grid with 26 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Blue is working with 2 squares while orange has 26 to spare, a gap of 24. Attack the cramped end and the roomy regions tend to resolve themselves.

Square count is only half the story — how a color is arranged counts too. Blue is the tidiest shape on #259, packing 100% of its 2×1 bounding box, whereas sea green straggles across its 3×4 box at just 58%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Here is the full footprint of every color on puzzle #259, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

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

Khaki occupies 2 squares within rows 8–9, columns 9–9. It sits wholly inside column 9, so its queen must be in that column.

Slate: 3 squares across rows 3–5, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Indigo covers 4 squares, spanning 2 rows and 3 columns (rows 1–2, columns 1–3).

Sea green occupies 7 squares within rows 8–10, columns 7–10.

Amber: 8 squares across rows 1–2, columns 3–7.

Pale yellow covers 11 squares, spanning 6 rows and 3 columns (rows 3–8, columns 6–8).

Mint occupies 18 squares within rows 2–6, columns 1–6.

Red: 19 squares across rows 1–8, columns 8–10.

Orange covers 26 squares, spanning 5 rows and 7 columns (rows 6–10, columns 1–7).

Where to start on this board

Open on the tightest color. Here that is blue, confined to 2 squares between rows 2 and 3 and columns 2 and 2. Its queen has nowhere else to go, which means those 2 rows and 1 column carry more constraint than anywhere else on #259.

Because blue stays inside column 2, its queen belongs to that column and no other region may use it. Cross out the rest of column 2 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.

Work outward to the other flat regions next. Indigo, amber and khaki 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 #259

A quick way to find the next move is to count how many different colors touch each line. On this board, row 10 is the narrowest, touched by only 2 colors. Column 10 is the tightest column, with 2 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 8 is the crowded extreme at 5 colors — worth revisiting later, not now.

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.

Two checks, run alternately, will carry you through the 100 squares of this grid. First: has any of the ten regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 10x10 boards almost always run one of those checks and forget the other.

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 #259 is rated hard

The rating reflects how much the board concedes up front. Puzzle #259 has three regions confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 24 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

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.

Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 10 on this board.

A non-spoiling distribution check: solved, #259 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 #259?

Puzzle #259 is rated hard. Its smallest color region holds 2 squares, 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 #259 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #259, 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 #259?

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