10x10 Queens Puzzle #164

Difficulty: easy. 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 19-step reasoning is written out below.

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Step-by-step solution for puzzle #164

What follows is a complete solve of puzzle #164, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 10 queen placements and 9 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 shadow, region confined to one line, pairing argument on rows, last square in a row, last square in a column.

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

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

  2. Step 2. Slate is down to 4 candidates in rows 5 and 6, columns 4–6, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about slate itself.

  3. Step 3. The red region has 2 squares left, spread across rows 6 and 7, columns 4 and 5. You do not need to know which of them takes the queen: every one of those options attacks the same 5 squares, so they can be crossed out now.

  4. Step 4. All of the slate region's remaining squares (2 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.

    Puzzle #164 after step 4: 1 of 10 queens placed
    The board after step 4 — 1 of 10 queens placed.

  5. Step 5. Orange cannot leave row 9 — every one of its open squares (3 open squares) sits there. That row belongs to orange, so every square on it owned by another colour is dead: 5 squares in all.

  6. Step 6. All of the khaki region's remaining squares (3 open squares) lie in row 8. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.

  7. Step 7. Wherever pale yellow 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 #164 after step 7: 1 of 10 queens placed
    The board after step 7 — 1 of 10 queens placed.

  8. Step 8. Here is the pairing argument. The red and pale yellow regions have no open squares outside rows 6 and 7 — 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.

  9. Step 9. The amber region has 5 squares left, spread across rows 1, 2, and 4, columns 8 and 9. 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.

  10. Step 10. Wherever mint ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.

  11. Step 11. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: slate has one square left, so its queen takes row 5, column 6; row 6 is down to one square, putting a queen on row 6, column 4; mint has one square left, so its queen takes row 4, column 3; orange has one square left, so its queen takes row 9, column 7; pale yellow has one square left, so its queen takes row 7, column 9; amber has one square left, so its queen takes row 2, column 8; column 2 is down to one square, putting a queen on row 1, column 2; blue has one square left, so its queen takes row 3, column 5; khaki has one square left, so its queen takes row 8, 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 8, column 1.

    Puzzle #164 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 #164

This is puzzle number 164 in the 10x10 collection, rated easy. 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 #164 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 #164 are: sea green (1 square), red (3 squares), slate (4 squares), orange (9 squares), mint (11 squares), khaki (12 squares), amber (14 squares), blue (14 squares), indigo (15 squares), pale yellow (17 squares). An even split would hand every color 10 squares, so the sea green region is running 9 squares under that average while the pale yellow region is the roomiest area on the grid with 17 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 pale yellow has 17 to spare, a gap of 16. 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 sea green is the most solid, filling 100% of its 1×1 box, and mint the most diffuse at 46% of its 6×4 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.

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

Red: 3 squares across rows 6–7, columns 4–5.

Slate covers 4 squares, spanning 2 rows and 3 columns (rows 5–6, columns 4–6).

Orange occupies 9 squares within rows 9–10, columns 3–8.

Mint: 11 squares across rows 3–8, columns 1–4.

Khaki covers 12 squares, spanning 4 rows and 5 columns (rows 7–10, columns 1–5).

Amber occupies 14 squares within rows 1–7, columns 8–10.

Blue: 14 squares across rows 1–4, columns 5–9.

Indigo covers 15 squares, spanning 5 rows and 5 columns (rows 1–5, columns 1–5).

Pale yellow occupies 17 squares within rows 5–10, columns 6–10.

Where to start on this board

Sea green is the natural opening on #164: it is the smallest region on the grid at 1 square, boxed into rows 10–10 and columns 10–10. 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. Slate, red and orange 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 #164

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

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 7, meeting 5 colors, is the opposite case — the busiest line on #164 and the last one you should expect to crack.

Working through the middle game

After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.

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 #164 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 #164 is rated easy

Ratings track how generous the opening is. On #164 you get two regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 16-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.

Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.

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.

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 #164 is solved, 3 queens sit on the outer edge of the grid and 7 sit inside it, with 1 queen 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 #164?

Puzzle #164 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #164 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #164, 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 #164?

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