10x10 Queens Puzzle #140

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.

Previous puzzle · All 10x10 puzzles · Next puzzle

Step-by-step solution for puzzle #140

What follows is a complete solve of puzzle #140, 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 rows.

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

Show the full walkthrough (11 steps) — contains the solution

  1. Step 1. Only row 10, column 6 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 6, the rest of the sea green region and every square adjacent to it — diagonals included.

  2. Step 2. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 9 squares come off.

  3. Step 3. Blue cannot leave column 10 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 8 squares in all.

  4. Step 4. Wherever slate ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

    Puzzle #140 after step 4: 1 of 10 queens placed
    The board after step 4 — 1 of 10 queens placed.

  5. Step 5. Khaki is down to 2 candidates in rows 8 and 9, columns 2 and 5, and they overlap enough to cover 4 squares elsewhere. Cross those out without deciding anything about khaki itself.

  6. Step 6. 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: 7 squares come off in one stroke.

  7. Step 7. Mint cannot leave row 4 — every one of its open squares (3 open squares) sits there. That row belongs to mint, so every square on it owned by another colour is dead: 6 squares in all.

    Puzzle #140 after step 7: 1 of 10 queens placed
    The board after step 7 — 1 of 10 queens placed.

  8. Step 8. The slate region is down to one open square, at row 5, column 9. Every colour holds exactly one queen, so that square is forced. Row 5 and column 9 are now spoken for, along with the squares touching that queen.

  9. Step 9. Pale yellow cannot leave column 7 — every one of its open squares (3 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 3 squares in all.

  10. Step 10. All of the orange region's remaining squares (2 open squares) lie in column 8. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 3 squares come off.

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

    Puzzle #140 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 #140

This is puzzle number 140 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.

There is exactly one way to finish #140, 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 #140 are: sea green (1 square), indigo (2 squares), blue (4 squares), slate (6 squares), pale yellow (8 squares), khaki (10 squares), mint (11 squares), amber (15 squares), orange (16 squares), red (27 squares). An even split would hand every color 10 squares, so the sea green region is running 9 squares under that average while the red region is the roomiest area on the grid with 27 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Sea green is working with 1 square while red has 27 to spare, a gap of 26. Attack the cramped end and the roomy regions tend to resolve themselves.

A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×2 box, and khaki the most diffuse at 56% of its 3×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

Region by region

Here is the full footprint of every color on puzzle #140, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Sea green occupies 1 square within rows 10–10, columns 6–6. It never leaves row 10, so its queen is locked to that row.

Indigo: 2 squares across rows 1–1, columns 1–2. It never leaves row 1, so its queen is locked to that row.

Blue covers 4 squares, spanning 3 rows and 2 columns (rows 1–3, columns 9–10).

Slate occupies 6 squares within rows 4–5, columns 7–10.

Pale yellow: 8 squares across rows 5–8, columns 6–8.

Khaki covers 10 squares, spanning 3 rows and 6 columns (rows 8–10, columns 1–6).

Mint occupies 11 squares within rows 2–4, columns 1–5.

Amber: 15 squares across rows 1–3, columns 3–9.

Orange covers 16 squares, spanning 5 rows and 4 columns (rows 6–10, columns 7–10).

Red occupies 27 squares within rows 3–9, columns 1–6.

Where to start on this board

Sea green is the natural opening on #140: it is the smallest region on the grid at 1 square, boxed into rows 10–10 and columns 6–6. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.

Because sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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 and slate 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 #140

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 4, 5, 6, 7, 9 and 10 are the narrowest, touched by only 3 colors. Column 10 is the tightest column, with 3 colors.

A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 9 rows. Row 8 is the crowded extreme at 4 colors — worth revisiting later, not now.

Working through the middle game

The middle of #140 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.

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.

Resist the plausible-looking queen. On #140 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 #140 is rated hard

The rating reflects how much the board concedes up front. Puzzle #140 has three regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 26 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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 #140 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.

A non-spoiling distribution check: solved, #140 carries 3 queens on the perimeter against 7 queens inside, 1 of them cornered. 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.

Full rules · Strategy library · Puzzle statistics

Questions about this puzzle

How hard is 10x10 Queens puzzle #140?

Puzzle #140 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #140 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #140, 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 #140?

Start with the sea green region — the most constrained color on this board, with 1 square spanning rows 10–10 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.