8x8 Queens Puzzle #16

Difficulty: hard. 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 9-step reasoning is written out below.

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Step-by-step solution for puzzle #16

What follows is a complete solve of puzzle #16, worked the way a person works it rather than the way a computer does. It runs to 9 deductions — 8 queen placements and 1 elimination step 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 confined to one line.

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

Show the full walkthrough (5 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. The blue region is down to one open square, at row 3, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 6, the remaining blue squares, and the ring of neighbours around row 3, column 6.

  3. Step 3. Only row 8, column 5 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 5, in khaki and in the ring around that square is now dead.

    Puzzle #16 after step 3: 3 of 8 queens placed
    The board after step 3 — 3 of 8 queens placed.

  4. Step 4. 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: 2 squares come off.

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

    Puzzle #16 after step 5: 8 of 8 queens placed
    The board after step 5 — 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 #16

This is puzzle number 16 in the 8x8 collection, rated hard. 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 #16 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 #16 are: indigo (1 square), khaki (1 square), slate (5 squares), amber (6 squares), blue (6 squares), mint (7 squares), pale yellow (11 squares), red (27 squares). An even split would hand every color 8 squares, so the indigo region is running 7 squares under that average while the red region is the roomiest area on the grid with 27 squares.

Every color takes one queen and one only, whatever its size — which is why the 26 square difference between indigo and red 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 56% of its 8×6 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

Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a hard board.

Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 7–7). It never leaves row 1, so its queen is locked to that row.

Khaki occupies 1 square within rows 8–8, columns 5–5. It never leaves row 8, so its queen is locked to that row.

Slate: 5 squares across rows 4–5, columns 1–3.

Amber covers 6 squares, spanning 3 rows and 2 columns (rows 1–3, columns 1–2).

Blue occupies 6 squares within rows 1–3, columns 6–8.

Mint: 7 squares across rows 1–4, columns 3–4.

Pale yellow covers 11 squares, spanning 4 rows and 4 columns (rows 5–8, columns 1–4).

Red occupies 27 squares within rows 1–8, columns 3–8.

Where to start on this board

The most constrained color is where a solve should begin, and on #16 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. Khaki 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 #16

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 6 and 7 are the narrowest, touched by only 2 colors. Columns 5, 6 and 8 are the tightest columns, with 2 colors.

A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 7 rows. Row 1 is the crowded extreme at 5 colors — worth revisiting later, not now.

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.

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 #16 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 #16 is rated hard

The hard label comes out of four measurable things about #16: four of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 26. Boards that score tight on those measures give up their first queen fast.

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.

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 #16 is row 6.

A non-spoiling distribution check: solved, #16 carries 4 queens on the perimeter against 4 queens inside, with the corners empty. 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.

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Questions about this puzzle

How hard is 8x8 Queens puzzle #16?

Puzzle #16 is rated hard. 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 8x8 puzzle #16 have only one solution?

Yes. Every puzzle in the 8x8 collection, including #16, 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 #16?

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.