10x10 Queens Puzzle #244

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 20-step reasoning is written out below.

Previous puzzle · All 10x10 puzzles · Next puzzle

Step-by-step solution for puzzle #244

What follows is a complete solve of puzzle #244, worked the way a person works it rather than the way a computer does. It runs to 20 deductions — 10 queen placements and 10 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, pairing argument on rows, last square in a region, region confined to one line, pairing argument on columns.

Show the full walkthrough (14 steps) — contains the solution

  1. Step 1. Wherever blue 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.

  2. Step 2. Mint is down to 4 candidates in rows 3 and 4, columns 5–7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.

  3. Step 3. The khaki region has 4 squares left, spread across rows 6–8, columns 1 and 2. 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 amber and blue regions have no open squares outside rows 2 and 3 — 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: 10 squares come off in one stroke.

  5. Step 5. Only row 4, column 7 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 7, the rest of the mint region and every square adjacent to it — diagonals included.

    Puzzle #244 after step 5: 1 of 10 queens placed
    The board after step 5 — 1 of 10 queens placed.

  6. Step 6. All of the indigo region's remaining squares (7 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 1 square come off.

  7. Step 7. Wherever red ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.

  8. Step 8. Khaki is down to 3 candidates in rows 6–8, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.

  9. Step 9. Amber, khaki, and sea green are now boxed into columns 1–3 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 9 squares at once.

    Puzzle #244 after step 9: 1 of 10 queens placed
    The board after step 9 — 1 of 10 queens placed.

  10. Step 10. The red region is down to one open square, at row 6, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 4, the remaining red squares, and the ring of neighbours around row 6, column 4.

  11. Step 11. Only row 9, column 5 is still open in orange, and since each colour must take a queen, that settles it. Everything else in row 9, in column 5, in orange and in the ring around that square is now dead.

  12. Step 12. All of the khaki region's remaining squares (2 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 3 squares come off.

  13. Step 13. Sea green cannot leave row 10 — every one of its open squares (2 open squares) sits there. That row belongs to sea green, so every square on it owned by another colour is dead: 3 squares in all.

  14. Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: pale yellow has one square left, so its queen takes row 7, column 6; indigo has one square left, so its queen takes row 1, column 8; blue has one square left, so its queen takes row 3, column 9; amber has one square left, so its queen takes row 2, column 3; khaki has one square left, so its queen takes row 8, column 2; slate has one square left, so its queen takes row 5, column 10; sea green has one square left, so its queen takes row 10, 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 10, column 1.

    Puzzle #244 after step 14: 10 of 10 queens placed
    The board after step 14 — 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 #244

This is puzzle number 244 in the 10x10 collection, rated hard. The grid holds 100 squares split between ten colored regions, and a finished board carries ten queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

Puzzle #244 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 #244 are: blue (3 squares), khaki (4 squares), amber (5 squares), mint (5 squares), orange (7 squares), sea green (7 squares), red (10 squares), indigo (14 squares), slate (19 squares), pale yellow (26 squares). An even split would hand every color 10 squares, so the blue region is running 7 squares under that average while the pale yellow region is the roomiest area on the grid with 26 squares.

Every color takes one queen and one only, whatever its size — which is why the 23 square difference between blue and pale yellow is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.

A region's outline can constrain it more than its size does. Here amber is the most solid, filling 83% of its 2×3 box, and slate the most diffuse at 53% of its 9×4 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

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.

Blue covers 3 squares, spanning 2 rows and 2 columns (rows 2–3, columns 8–9).

Khaki occupies 4 squares within rows 6–8, columns 1–2.

Amber: 5 squares across rows 2–3, columns 1–3.

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

Orange occupies 7 squares within rows 7–10, columns 3–5.

Sea green: 7 squares across rows 7–10, columns 1–3.

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

Indigo occupies 14 squares within rows 1–2, columns 1–9.

Slate: 19 squares across rows 1–9, columns 7–10.

Pale yellow covers 26 squares, spanning 7 rows and 7 columns (rows 4–10, columns 4–10).

Where to start on this board

Blue is the natural opening on #244: it is the smallest region on the grid at 3 squares, boxed into rows 2–3 and columns 8–9. Since one of those squares must hold a queen, that block of 2 rows and 2 columns is the most contested part of the board.

Blue does not collapse onto a single line, so it will not hand you a queen on move one. What it does give you is a short list of 3 candidates. Take each one in turn, imagine the queen there, cross out its row, its column and the eight squares around it, and ask whether any other region is left with nowhere to go. One or two candidates usually fail that test immediately.

Work outward to the other flat regions next. Indigo, amber and mint 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 #244

A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Column 10 is the tightest column, with 2 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 8 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.

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 #244 is rated hard

Ratings track how generous the opening is. On #244 you get no region pinned to a single row or column, a smallest color of 3 squares, a tightest row crossed by 2 colors, and a 23-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.

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.

There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #244 is row 1.

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.

Full rules · Strategy library · Puzzle statistics

Questions about this puzzle

How hard is 10x10 Queens puzzle #244?

Puzzle #244 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 2 colors, and no region is confined to a single row or column, so the opening move takes some hunting.

Does 10x10 puzzle #244 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #244, 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 #244?

Start with the blue region — the most constrained color on this board, with 3 squares spanning rows 2–3 and columns 8–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.