10x10 Queens Puzzle #139

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 #139

What follows is a complete solve of puzzle #139, 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 region shadow.

Techniques this board needs: last square in a region, region confined to one line, region shadow.

Show the full walkthrough (8 steps) — contains the solution

  1. Step 1. Only row 9, column 10 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 10, the rest of the orange region and every square adjacent to it — diagonals included.

  2. Step 2. All of the mint region's remaining squares (3 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.

  3. Step 3. The slate region has 7 squares left, spread across rows 5, 6, 7, 8, and 10, columns 6–8. 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.

  4. Step 4. Wherever red ends up among its 5 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.

    Puzzle #139 after step 4: 1 of 10 queens placed
    The board after step 4 — 1 of 10 queens placed.

  5. Step 5. Only row 10, column 8 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 8, the rest of the sea green region and every square adjacent to it — diagonals included.

  6. Step 6. All of the red 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: 4 squares come off.

  7. Step 7. Amber cannot leave column 7 — every one of its open squares (3 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 5 squares in all.

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

    Puzzle #139 after step 8: 10 of 10 queens placed
    The board after step 8 — 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 #139

10x10 puzzle #139 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.

There is exactly one way to finish #139, 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 #139 are: orange (1 square), mint (3 squares), sea green (6 squares), slate (8 squares), khaki (9 squares), indigo (11 squares), red (11 squares), amber (15 squares), pale yellow (16 squares), blue (20 squares). An even split would hand every color 10 squares, so the orange region is running 9 squares under that average while the blue region is the roomiest area on the grid with 20 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Orange is working with 1 square while blue has 20 to spare, a gap of 19. 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. Mint is the tidiest shape on #139, packing 100% of its 3×1 bounding box, whereas slate straggles across its 6×3 box at just 44%. 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.

Orange covers 1 square, spanning 1 row and 1 column (rows 9–9, columns 10–10). It never leaves row 9, so its queen is locked to that row.

Mint occupies 3 squares within rows 4–6, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Sea green: 6 squares across rows 7–10, columns 8–10.

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

Khaki occupies 9 squares within rows 7–10, columns 4–6.

Indigo: 11 squares across rows 1–3, columns 1–6.

Red covers 11 squares, spanning 6 rows and 3 columns (rows 4–9, columns 8–10).

Amber occupies 15 squares within rows 1–4, columns 7–10.

Pale yellow: 16 squares across rows 3–10, columns 1–3.

Blue covers 20 squares, spanning 7 rows and 4 columns (rows 2–8, columns 3–6).

Where to start on this board

Open on the tightest color. Here that is orange, confined to 1 square between rows 9 and 9 and columns 10 and 10. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #139.

Because orange never leaves row 9, its queen must be in that row — and that instantly settles row 9 for everyone else. Sweep along it and X out every square that is not orange. 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.

Nothing on this grid is pinned to one or two lines, which is why the opening takes patience. Scan the columns for one where only two or three colors appear — whichever region is forced into it claims that column outright. Chaining two of those observations is normally enough to break #139 open.

The tightest lines on puzzle #139

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. Columns 2 and 7 are the tightest columns, with 2 colors.

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 9, meeting 6 colors, is the opposite case — the busiest line on #139 and the last one you should expect to crack.

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.

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 #139 is rated hard

Ratings track how generous the opening is. On #139 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 19-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.

The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a hard board from the rest.

Stuck on #139 despite the hard 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 1 on this board.

Here is a shape check that gives nothing away: when #139 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 #139?

Puzzle #139 is rated hard. Its smallest color region holds 1 square, 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 #139 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #139, 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 #139?

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