10x10 Queens Puzzle #295

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 18-step reasoning is written out below.

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Step-by-step solution for puzzle #295

What follows is a complete solve of puzzle #295, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 10 queen placements and 8 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: region shadow, region confined to one line, pairing argument on rows, last square in a region, line confined to one region, pairing argument on columns, last square in a row.

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

  1. Step 1. Wherever indigo 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.

  2. Step 2. All of the orange 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: 10 squares come off.

  3. Step 3. The amber region has 4 squares left, spread across rows 1 and 2, columns 2–4. 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.

    Puzzle #295 after step 3: 0 of 10 queens placed
    The board after step 3 — 0 of 10 queens placed.

  4. Step 4. 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.

  5. Step 5. Only row 3, column 8 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 8, the rest of the blue region and every square adjacent to it — diagonals included.

  6. Step 6. The mint region has 3 squares left, spread across rows 4 and 5, columns 4 and 5. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.

  7. Step 7. Column 10 has only pale yellow squares left (7 open squares). The column needs a queen and only pale yellow can provide one, which pins that colour to this column and kills its squares elsewhere: 12 squares.

    Puzzle #295 after step 7: 1 of 10 queens placed
    The board after step 7 — 1 of 10 queens placed.

  8. Step 8. Here is the pairing argument. The amber, mint, slate, and sea green regions have no open squares outside columns 2–5 — 4 colours confined to 4 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 8 squares come off in one stroke.

  9. Step 9. Red cannot leave column 6 — every one of its open squares (3 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. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: row 2 is down to one square, putting a queen on row 2, column 2; slate has one square left, so its queen takes row 5, column 3; mint has one square left, so its queen takes row 4, column 5; red has one square left, so its queen takes row 6, column 6; row 7 is down to one square, putting a queen on row 7, column 10; khaki has one square left, so its queen takes row 8, column 7; indigo has one square left, so its queen takes row 1, column 9; sea green has one square left, so its queen takes row 10, column 4; orange has one square left, so its queen takes row 9, column 1. 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 9, column 1.

    Puzzle #295 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 #295

10x10 puzzle #295 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 #295 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 #295 are: orange (2 squares), indigo (5 squares), amber (6 squares), blue (6 squares), red (9 squares), khaki (9 squares), sea green (10 squares), mint (11 squares), slate (14 squares), pale yellow (28 squares). An even split would hand every color 10 squares, so the orange region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 28 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Orange is working with 2 squares while pale yellow has 28 to spare, a gap of 26. 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. Orange is the tidiest shape on #295, packing 100% of its 2×1 bounding box, whereas khaki straggles across its 3×6 box at just 50%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Below is every color on #295 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.

Orange covers 2 squares, spanning 2 rows and 1 column (rows 9–10, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Indigo occupies 5 squares within rows 1–2, columns 6–9.

Amber: 6 squares across rows 1–2, columns 1–4.

Blue covers 6 squares, spanning 3 rows and 4 columns (rows 1–3, columns 7–10).

Red occupies 9 squares within rows 4–7, columns 3–6.

Khaki: 9 squares across rows 8–10, columns 4–9.

Sea green covers 10 squares, spanning 4 rows and 4 columns (rows 7–10, columns 2–5).

Mint occupies 11 squares within rows 1–5, columns 4–7.

Slate: 14 squares across rows 2–8, columns 1–3.

Pale yellow covers 28 squares, spanning 8 rows and 6 columns (rows 3–10, columns 5–10).

Where to start on this board

Open on the tightest color. Here that is orange, confined to 2 squares between rows 9 and 10 and columns 1 and 1. Its queen has nowhere else to go, which means those 2 rows and 1 column carry more constraint than anywhere else on #295.

Because orange stays inside column 1, its queen belongs to that column and no other region may use it. Cross out the rest of column 1 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. Amber and indigo 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 #295

A quick way to find the next move is to count how many different colors touch each line. On this board, row 6 is the narrowest, touched by only 3 colors. Column 10 is the tightest column, with 2 colors.

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 2, meeting 5 colors, is the opposite case — the busiest line on #295 and the last one you should expect to crack.

Working through the middle game

Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.

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.

Resist the plausible-looking queen. On #295 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 #295 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 3 colors, and a region-size spread of 26. 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.

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 column 10 on this board.

Here is a shape check that gives nothing away: when #295 is solved, 4 queens sit on the outer edge of the grid and 6 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.

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 #295?

Puzzle #295 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #295 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #295, 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 #295?

Start with the orange region — the most constrained color on this board, with 2 squares spanning rows 9–10 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.