10x10 Queens Puzzle #43

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 #43

What follows is a complete solve of puzzle #43, 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 (11 steps) — contains the solution

  1. Step 1. Only row 4, column 5 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 5, the rest of the mint region and every square adjacent to it — diagonals included.

  2. Step 2. The sea green region is down to one open square, at row 10, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 9, the remaining sea green squares, and the ring of neighbours around row 10, column 9.

  3. Step 3. 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: 6 squares in all.

  4. Step 4. All of the red 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.

    Puzzle #43 after step 4: 2 of 10 queens placed
    The board after step 4 — 2 of 10 queens placed.

  5. Step 5. Pale yellow cannot leave column 8 — 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: 6 squares in all.

  6. Step 6. The orange region has 2 squares left, spread across rows 8 and 9, columns 7 and 10. 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. Only row 7, column 8 is still open in pale yellow, and since each colour must take a queen, that settles it. Everything else in row 7, in column 8, in pale yellow and in the ring around that square is now dead.

    Puzzle #43 after step 7: 3 of 10 queens placed
    The board after step 7 — 3 of 10 queens placed.

  8. Step 8. The red region is down to one open square, at row 6, column 1. Every colour holds exactly one queen, so that square is forced. Row 6 and column 1 are now spoken for, along with the squares touching that queen.

  9. Step 9. The blue region has 4 squares left, spread across rows 2 and 3, 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.

  10. Step 10. All of the slate 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: 1 square come off.

  11. Step 11. 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: orange has one square left, so its queen takes row 9, column 7; amber has one square left, so its queen takes row 2, column 6; indigo has one square left, so its queen takes row 1, column 4; blue has one square left, so its queen takes row 3, column 2; slate has one square left, so its queen takes row 5, column 10; khaki has one square left, so its queen takes row 8, column 3. 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 8, column 3.

    Puzzle #43 after step 11: 10 of 10 queens placed
    The board after step 11 — 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 #43

10x10 puzzle #43 is rated medium. 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.

There is exactly one way to finish #43, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these ten regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #43 are: mint (1 square), sea green (1 square), red (2 squares), indigo (3 squares), pale yellow (5 squares), orange (10 squares), amber (12 squares), blue (15 squares), slate (21 squares), khaki (30 squares). An even split would hand every color 10 squares, so the mint region is running 9 squares under that average while the khaki region is the roomiest area on the grid with 30 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Mint is working with 1 square while khaki has 30 to spare, a gap of 29. 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. Indigo is the tidiest shape on #43, packing 100% of its 1×3 bounding box, whereas amber straggles across its 4×6 box at just 50%. 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.

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

Sea green occupies 1 square within rows 10–10, columns 9–9. It never leaves row 10, so its queen is locked to that row.

Red: 2 squares across rows 6–7, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Indigo covers 3 squares, spanning 1 row and 3 columns (rows 1–1, columns 4–6). It never leaves row 1, so its queen is locked to that row.

Pale yellow occupies 5 squares within rows 6–8, columns 8–9.

Orange: 10 squares across rows 8–10, columns 6–10.

Amber covers 12 squares, spanning 4 rows and 6 columns (rows 2–5, columns 3–8).

Blue occupies 15 squares within rows 1–5, columns 1–5.

Slate: 21 squares across rows 1–7, columns 6–10.

Khaki covers 30 squares, spanning 6 rows and 7 columns (rows 5–10, columns 1–7).

Where to start on this board

Open on the tightest color. Here that is mint, confined to 1 square between rows 4 and 4 and columns 5 and 5. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #43.

Because mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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, sea green and red 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 #43

A quick way to find the next move is to count how many different colors touch each line. On this board, row 9 is the narrowest, touched by only 2 colors. Columns 2 and 10 are the tightest columns, 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 7 is the crowded extreme at 4 colors — worth revisiting later, not now.

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.

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 #43 is rated medium

Ratings track how generous the opening is. On #43 you get six regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 29-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.

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.

Stuck on #43 despite the medium rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.

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 9 on this board.

A non-spoiling distribution check: solved, #43 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 #43?

Puzzle #43 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and six of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #43 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #43, 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 #43?

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