8x8 Queens Puzzle #20

Difficulty: easy. 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 #20

What follows is a complete solve of puzzle #20, 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 line confined to one region.

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

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

  1. Step 1. Only row 1, column 1 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 1, 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. Amber cannot leave row 2 — every one of its open squares (2 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 1 square in all.

  4. Step 4. All of the slate region's remaining squares (2 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 3 squares come off.

    Puzzle #20 after step 4: 2 of 8 queens placed
    The board after step 4 — 2 of 8 queens placed.

  5. Step 5. Red cannot leave column 5 — every one of its open squares (2 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 4 squares in all.

  6. Step 6. The khaki region has 3 squares left, spread across rows 7 and 8, columns 7 and 8. 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.

  7. Step 7. Row 4 has only mint squares left (3 open squares). The row needs a queen and only mint can provide one, which pins that colour to this row and kills its squares elsewhere: 4 squares.

  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: column 7 is down to one square, putting a queen on row 8, column 7; slate has one square left, so its queen takes row 5, column 2; red has one square left, so its queen takes row 6, column 5; pale yellow has one square left, so its queen takes row 7, column 3; amber has one square left, so its queen takes row 2, column 4; mint has one square left, so its queen takes row 4, 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 4, column 8.

    Puzzle #20 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 #20

This is puzzle number 20 in the 8x8 collection, rated easy. 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 #20 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 #20 are: indigo (1 square), blue (1 square), red (2 squares), slate (6 squares), khaki (6 squares), pale yellow (10 squares), amber (15 squares), mint (23 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 23 squares.

Every color takes one queen and one only, whatever its size — which is why the 22 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.

Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #20, packing 100% of its 1×1 bounding box, whereas mint straggles across its 7×8 box at just 41%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

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 easy board.

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

Blue: 1 square across rows 3–3, columns 6–6. It never leaves row 3, so its queen is locked to that row.

Red covers 2 squares, spanning 2 rows and 1 column (rows 5–6, columns 5–5). It sits wholly inside column 5, so its queen must be in that column.

Slate occupies 6 squares within rows 5–8, columns 1–2.

Khaki: 6 squares across rows 7–8, columns 5–8.

Pale yellow covers 10 squares, spanning 4 rows and 4 columns (rows 5–8, columns 2–5).

Amber occupies 15 squares within rows 1–3, columns 1–6.

Mint: 23 squares across rows 1–7, columns 1–8.

Where to start on this board

The most constrained color is where a solve should begin, and on #20 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 1 to column 1 — 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. Blue, red 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 #20

A quick way to find the next move is to count how many different colors touch each line. On this board, row 4 is the narrowest, touched by only 1 color. Columns 7 and 8 are the tightest columns, with 2 colors.

A line with few colors is a line close to resolving. Where only 1 color appear, one of them owns the row outright and the others must fit into the remaining 7 rows. Row 7 is the crowded extreme at 4 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.

With eight queens to seat and eight regions competing over 64 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 8x8 boards quickly and players who stall halfway.

The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.

Why puzzle #20 is rated easy

The easy label comes out of four measurable things about #20: five of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 1 color, and its largest and smallest regions differ by 22. 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 easy 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 #20 is row 4.

A non-spoiling distribution check: solved, #20 carries 3 queens on the perimeter against 5 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.

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Questions about this puzzle

How hard is 8x8 Queens puzzle #20?

Puzzle #20 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 1 color, and five of its regions are confined to a single row or column, which gives you a forced move early.

Does 8x8 puzzle #20 have only one solution?

Yes. Every puzzle in the 8x8 collection, including #20, 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 #20?

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