8x8 Queens Puzzle #164

Difficulty: medium. Place 8 queens on this 8x8 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 #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 13 deductions — 8 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 pairing argument on columns.

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

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

  1. Step 1. Only row 1, column 7 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 7, the rest of the indigo region and every square adjacent to it — diagonals included.

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

  3. Step 3. Slate cannot leave column 8 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 4 squares in all.

  4. Step 4. The khaki region is down to one open square, at row 8, column 6. Every colour holds exactly one queen, so that square is forced. Row 8 and column 6 are now spoken for, along with the squares touching that queen.

    Puzzle #164 after step 4: 2 of 8 queens placed
    The board after step 4 — 2 of 8 queens placed.

  5. Step 5. Blue, mint, and slate are now boxed into rows 3–5 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 3 squares at once.

  6. Step 6. The red region has 4 squares left, spread across rows 6 and 7, columns 3–5. 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.

  7. Step 7. Amber and pale yellow are now boxed into columns 1 and 2 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 4 squares at once.

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

    Puzzle #164 after step 8: 8 of 8 queens placed
    The board after step 8 — 8 of 8 queens placed.

That completes the board: 8 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 8x8 Queens puzzle #164

This is puzzle number 164 in the 8x8 collection, rated medium. The grid holds 64 squares split between eight colored regions, and a finished board carries eight 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: indigo (1 square), slate (2 squares), amber (5 squares), khaki (5 squares), blue (7 squares), pale yellow (9 squares), red (17 squares), mint (18 squares). An even split would hand every color 8 squares, so the indigo region is running 7 squares under that average while the mint region is the roomiest area on the grid with 18 squares.

Every color takes one queen and one only, whatever its size — which is why the 17 square difference between indigo and mint 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.

Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×1 box that encloses it, while red is the most scattered at 47% of its 6×6 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.

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.

Indigo occupies 1 square within rows 1–1, columns 7–7. It never leaves row 1, so its queen is locked to that row.

Slate: 2 squares across rows 4–5, columns 8–8. It sits wholly inside column 8, so its queen must be in that column.

Amber covers 5 squares, spanning 2 rows and 3 columns (rows 1–2, columns 1–3).

Khaki occupies 5 squares within rows 7–8, columns 6–8.

Blue: 7 squares across rows 3–5, columns 1–3.

Pale yellow covers 9 squares, spanning 3 rows and 4 columns (rows 6–8, columns 1–4).

Red occupies 17 squares within rows 3–8, columns 3–8.

Mint: 18 squares across rows 1–4, columns 3–8.

Where to start on this board

The most constrained color is where a solve should begin, and on #164 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 7 to column 7 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.

Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Amber, slate and khaki 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 2 and 6 are the narrowest, touched by only 2 colors. Column 5 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 7 rows. Row 4 sits at the far end with 4 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 64 squares of this grid. First: has any of the eight regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 8x8 boards almost always run one of those checks and forget the other.

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 medium

Ratings track how generous the opening is. On #164 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 17-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.

Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.

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 8x8

At 8x8 the whole grid stays within a single glance, which makes systematic scanning practical. Read the rows in order and count the colors in each — a row touched by two colors is far more constrained than one touched by five, and that is usually where the next forced move hides. The board is small enough that re-scanning from the top after each placement costs almost nothing.

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 #164 is row 2.

Use this as a rough check on your progress: the completed board has 4 queens around the border and 4 queens in the middle, and nothing in the corners. 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.

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Questions about this puzzle

How hard is 8x8 Queens puzzle #164?

Puzzle #164 is rated medium. 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 8x8 puzzle #164 have only one solution?

Yes. Every puzzle in the 8x8 collection, including #164, has been verified to have exactly one valid arrangement of eight 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 indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.