10x10 Queens Puzzle #243

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 #243

What follows is a complete solve of puzzle #243, 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 pairing argument on columns.

Techniques this board needs: last square in a region, region shadow, pairing argument on rows, pairing argument on columns, region confined to one line.

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

  1. Step 1. Only row 5, column 8 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 8, the rest of the slate region and every square adjacent to it — diagonals included.

  2. Step 2. The red region is down to one open square, at row 6, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 3, the remaining red squares, and the ring of neighbours around row 6, column 3.

  3. Step 3. The sea green region has 3 squares left, spread across rows 9 and 10, 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.

  4. Step 4. Here is the pairing argument. The indigo, amber, blue, and mint regions have no open squares outside rows 1–4 — 4 colours confined to 4 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 3 squares come off in one stroke.

    Puzzle #243 after step 4: 2 of 10 queens placed
    The board after step 4 — 2 of 10 queens placed.

  5. Step 5. Indigo, blue, and khaki are now boxed into columns 1, 2, and 4 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 4 squares at once.

  6. Step 6. All of the sea green region's remaining squares (2 open squares) lie in column 5. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 7 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: mint has one square left, so its queen takes row 4, column 6; blue has one square left, so its queen takes row 3, column 2; indigo has one square left, so its queen takes row 1, column 1; khaki has one square left, so its queen takes row 8, column 4; sea green has one square left, so its queen takes row 10, column 5; 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 9; pale yellow has one square left, so its queen takes row 7, column 10. 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 7, column 10.

    Puzzle #243 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 #243

10x10 puzzle #243 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.

Like every board on this site, #243 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 #243 are: slate (1 square), red (1 square), sea green (4 squares), indigo (7 squares), blue (10 squares), orange (12 squares), amber (13 squares), mint (13 squares), khaki (18 squares), pale yellow (21 squares). An even split would hand every color 10 squares, so the slate region is running 9 squares under that average while the pale yellow region is the roomiest area on the grid with 21 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Slate is working with 1 square while pale yellow has 21 to spare, a gap of 20. Attack the cramped end and the roomy regions tend to resolve themselves.

Look at arrangement as well as area. Slate is the densest region on this grid at 100% of a 1×1 frame; mint is the loosest, occupying only 52% of its 5×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

Region by region

Below is every color on #243 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.

Slate: 1 square across rows 5–5, columns 8–8. It never leaves row 5, so its queen is locked to that row.

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

Sea green occupies 4 squares within rows 9–10, columns 3–5.

Indigo: 7 squares across rows 1–3, columns 1–3.

Blue covers 10 squares, spanning 3 rows and 4 columns (rows 3–5, columns 1–4).

Orange occupies 12 squares within rows 6–10, columns 5–8.

Amber: 13 squares across rows 1–3, columns 6–10.

Mint covers 13 squares, spanning 5 rows and 5 columns (rows 1–5, columns 4–8).

Khaki occupies 18 squares within rows 6–10, columns 1–5.

Pale yellow: 21 squares across rows 2–10, columns 7–10.

Where to start on this board

Open on the tightest color. Here that is slate, confined to 1 square between rows 5 and 5 and columns 8 and 8. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #243.

Because slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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. Red and sea green 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 #243

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 4, 7 and 8 are the narrowest, touched by only 3 colors. Columns 9 and 10 are the tightest columns, with 2 colors.

Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 3 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.

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 #243 is rated hard

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 3 colors, and a region-size spread of 20. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

If #243 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.

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 9 on this board.

Use this as a rough check on your progress: the completed board has 3 queens around the border and 7 queens in the middle, including 1 in a corner. 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 #243?

Puzzle #243 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #243 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #243, 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 #243?

Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.