8x8 Queens Puzzle #45
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 13-step reasoning is written out below.
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Step-by-step solution for puzzle #45
What follows is a complete solve of puzzle #45, 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 row.
Show the full walkthrough (9 steps) — contains the solution
Step 1. Only row 1, column 2 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 2, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. 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.
Step 3. Amber cannot leave row 2 — every one of its open squares (3 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 2 squares in all.
The board after step 3 — 2 of 8 queens placed. Step 4. All of the mint region's remaining squares (2 open squares) lie in column 3. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.
Step 5. Only row 7, column 4 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 4, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 6. 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: 1 square come off.
The board after step 6 — 3 of 8 queens placed. Step 7. Wherever red ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 8. Every open square left in row 8 (3 open squares) belongs to the khaki region. Since that row must contain a queen, khaki is the colour that supplies it — so khaki squares anywhere else on the board are out, 3 squares in all.
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: row 6 is down to one square, putting a queen on row 6, column 6; amber has one square left, so its queen takes row 2, column 8; khaki has one square left, so its queen takes row 8, column 7; blue has one square left, so its queen takes row 3, column 5; mint has one square left, so its queen takes row 4, column 3. 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 3.
The board after step 9 — 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 #45
Puzzle #45 is one of the hard-rated boards in the 8x8 set. Eight colors divide its 64 squares, and you win by placing eight queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
There is exactly one way to finish #45, 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 eight regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #45 are: indigo (1 square), slate (1 square), pale yellow (7 squares), amber (8 squares), blue (9 squares), khaki (11 squares), red (13 squares), mint (14 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 14 squares.
Every color takes one queen and one only, whatever its size — which is why the 13 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 #45, packing 100% of its 1×1 bounding box, whereas red straggles across its 5×6 box at just 43%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Below is every color on #45 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Indigo: 1 square across rows 1–1, columns 2–2. It never leaves row 1, so its queen is locked to that row.
Slate covers 1 square, spanning 1 row and 1 column (rows 5–5, columns 1–1). It never leaves row 5, so its queen is locked to that row.
Pale yellow occupies 7 squares within rows 7–8, columns 1–4.
Amber: 8 squares across rows 1–2, columns 4–8.
Blue covers 9 squares, spanning 3 rows and 5 columns (rows 2–4, columns 4–8).
Khaki occupies 11 squares within rows 4–8, columns 6–8.
Red: 13 squares across rows 4–8, columns 1–6.
Mint covers 14 squares, spanning 5 rows and 4 columns (rows 1–5, columns 1–4).
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 2 and 2. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #45.
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. Slate, amber and pale yellow 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 #45
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 6 are the narrowest, touched by only 2 colors. Columns 3, 5, 7 and 8 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 7 rows. Row 5, meeting 4 colors, is the opposite case — the busiest line on #45 and the last one you should expect to crack.
Working through the middle game
The middle of #45 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.
The whole middle game reduces to two sweeps. Sweep the eight colors looking for one with a single square left; sweep the eight rows and eight columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 8x8 solve from a stalled one.
A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.
Why puzzle #45 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers four single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 13. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
Stuck on #45 despite the hard 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 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 #45 is row 3.
Here is a shape check that gives nothing away: when #45 is solved, 4 queens sit on the outer edge of the grid and 4 sit inside it, and no queen lands 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
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- 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.
- 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 #45?
Puzzle #45 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 #45 have only one solution?
Yes. Every puzzle in the 8x8 collection, including #45, 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 #45?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.