10x10 Queens Puzzle #72

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 13-step reasoning is written out below.

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Step-by-step solution for puzzle #72

What follows is a complete solve of puzzle #72, worked the way a person works it rather than the way a computer does. It runs to 13 deductions — 10 queen placements and 3 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 (9 steps) — contains the solution

  1. Step 1. Only row 8, column 4 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 4, the rest of the khaki 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 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 3, the remaining sea green squares, and the ring of neighbours around row 10, column 3.

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

    Puzzle #72 after step 3: 2 of 10 queens placed
    The board after step 3 — 2 of 10 queens placed.

  4. Step 4. All of the slate 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: 8 squares come off.

  5. Step 5. Only row 7, column 2 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 2, the rest of the pale yellow region and every square adjacent to it — diagonals included.

  6. Step 6. The slate region is down to one open square, at row 5, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 1, the remaining slate squares, and the ring of neighbours around row 5, column 1.

    Puzzle #72 after step 6: 4 of 10 queens placed
    The board after step 6 — 4 of 10 queens placed.

  7. Step 7. Only row 4, column 10 is still open in mint, and since each colour must take a queen, that settles it. Everything else in row 4, in column 10, in mint and in the ring around that square is now dead.

  8. Step 8. Indigo is down to 4 candidates in rows 1–3, columns 5 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.

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

    Puzzle #72 after step 9: 10 of 10 queens placed
    The board after step 9 — 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 #72

This is puzzle number 72 in the 10x10 collection, rated medium. 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 #72, 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 #72 are: khaki (1 square), mint (2 squares), slate (2 squares), sea green (5 squares), amber (12 squares), pale yellow (12 squares), indigo (13 squares), blue (15 squares), red (18 squares), orange (20 squares). An even split would hand every color 10 squares, so the khaki region is running 9 squares under that average while the orange 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. Khaki is working with 1 square while orange has 20 to spare, a gap of 19. Attack the cramped end and the roomy regions tend to resolve themselves.

Look at arrangement as well as area. Mint is the densest region on this grid at 100% of a 2×1 frame; red is the loosest, occupying only 38% of its 6×8 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

Region by region

Here is the full footprint of every color on puzzle #72, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Khaki: 1 square across rows 8–8, columns 4–4. It never leaves row 8, so its queen is locked to that row.

Mint covers 2 squares, spanning 2 rows and 1 column (rows 4–5, columns 10–10). It sits wholly inside column 10, so its queen must be in that column.

Slate occupies 2 squares within rows 5–6, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Sea green: 5 squares across rows 8–10, columns 3–5.

Amber covers 12 squares, spanning 4 rows and 4 columns (rows 1–4, columns 7–10).

Pale yellow occupies 12 squares within rows 7–10, columns 1–5.

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

Blue covers 15 squares, spanning 5 rows and 5 columns (rows 2–6, columns 1–5).

Red occupies 18 squares within rows 2–7, columns 3–10.

Orange: 20 squares across rows 6–10, columns 4–10.

Where to start on this board

Khaki is the natural opening on #72: it is the smallest region on the grid at 1 square, boxed into rows 8–8 and columns 4–4. 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 khaki never leaves row 8, its queen must be in that row — and that instantly settles row 8 for everyone else. Sweep along it and X out every square that is not khaki. 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. Mint 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 #72

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, 6, 7, 8 and 9 are the tightest columns, with 3 colors.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 9 rows. By contrast row 8 meets 4 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

Working through the middle game

The middle of #72 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.

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.

Resist the plausible-looking queen. On #72 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 #72 is rated medium

The rating reflects how much the board concedes up front. Puzzle #72 has four regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 19 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

Stuck on #72 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 1 on this board.

As a sanity check rather than a spoiler: the finished solution to #72 puts 4 queens on the outer border and 6 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.

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 #72?

Puzzle #72 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #72 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #72, 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 #72?

Start with the khaki region — the most constrained color on this board, with 1 square spanning rows 8–8 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.