7x7 Queens Puzzle #39
Difficulty: easy. Place 7 queens on this 7x7 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 15-step reasoning is written out below.
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Step-by-step solution for puzzle #39
What follows is a complete solve of puzzle #39, worked the way a person works it rather than the way a computer does. It runs to 15 deductions — 7 queen placements and 8 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 proof by contradiction.
Techniques this board needs: last square in a region, region shadow, region confined to one line, last square in a column, proof by contradiction.
Show the full walkthrough (12 steps) — contains the solution
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.
Step 2. Blue is down to 3 candidates in rows 3 and 4, columns 6 and 7, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about blue itself.
Step 3. The slate region has 3 squares left, spread across rows 5 and 6, columns 6 and 7. 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.
Step 4. All of the pale yellow region's remaining squares (2 open squares) lie in row 7. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.
The board after step 4 — 1 of 7 queens placed. Step 5. Red cannot leave row 6 — every one of its open squares (2 open squares) sits there. That row belongs to red, so every square on it owned by another colour is dead: 3 squares in all.
Step 6. The mint region has 3 squares left, spread across rows 4 and 5, 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.
Step 7. Slate cannot leave row 5 — every one of its open squares (2 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 3 squares in all.
Step 8. All of the blue region's remaining squares (2 open squares) lie in row 3. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.
The board after step 8 — 1 of 7 queens placed. Step 9. Scan column 2 and only row 7, column 2 remains — a pale yellow square. Every column takes a queen, so it takes this one. Row 7 and column 2 are now spoken for, along with the squares touching that queen.
Step 10. Column 3 now has exactly one open square, row 2, column 3, in the amber region. Columns must each hold a queen, so that placement is forced. That clears row 2, column 3, the rest of the amber region and every square adjacent to it — diagonals included.
Step 11. Test row 3, column 7 for blue: place a queen there, run the forced consequences through, and column 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 12. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: blue has one square left, so its queen takes row 3, column 6; 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 7; red has one square left, so its queen takes row 6, column 5. 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 5.
The board after step 12 — 7 of 7 queens placed.
That completes the board: 7 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 7x7 Queens puzzle #39
7x7 puzzle #39 is rated easy. Seven colored regions cover its 49 squares, and your job is to seat seven queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Like every board on this site, #39 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.
How the colors divide this board
From smallest to largest, the regions on puzzle #39 are: indigo (1 square), blue (3 squares), slate (4 squares), mint (6 squares), pale yellow (6 squares), red (7 squares), amber (22 squares). An even split would hand every color 7 squares, so the indigo region is running 6 squares under that average while the amber region is the roomiest area on the grid with 22 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 1 square while amber has 22 to spare, a gap of 21. 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 indigo is the most solid, filling 100% of its 1×1 box, and pale yellow the most diffuse at 50% of its 4×3 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Here is the full footprint of every color on puzzle #39, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Indigo: 1 square across rows 1–1, columns 1–1. It never leaves row 1, so its queen is locked to that row.
Blue covers 3 squares, spanning 2 rows and 2 columns (rows 3–4, columns 6–7).
Slate occupies 4 squares within rows 4–6, columns 6–7.
Mint: 6 squares across rows 3–5, columns 3–5.
Pale yellow covers 6 squares, spanning 4 rows and 3 columns (rows 4–7, columns 1–3).
Red occupies 7 squares within rows 6–7, columns 4–7.
Amber: 22 squares across rows 1–6, columns 1–7.
Where to start on this board
The most constrained color is where a solve should begin, and on #39 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 and red 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 #39
A quick way to find the next move is to count how many different colors touch each line. On this board, row 2 is the narrowest, touched by only 1 color. Column 2 is the tightest column, 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 6 rows. Row 4 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
Keep alternating between two views of the board. In the region view you ask which of the seven colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 7x7 board unwinds.
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 #39 is rated easy
The rating reflects how much the board concedes up front. Puzzle #39 has two regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 1 color, and the gap between its largest and smallest regions is 21 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
If #39 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.
Techniques that work well at 7x7
At 7x7 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.
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 2 on this board.
A non-spoiling distribution check: solved, #39 carries 3 queens on the perimeter against 4 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
- 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 7x7 Queens puzzle #39?
Puzzle #39 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 1 color, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 7x7 puzzle #39 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #39, has been verified to have exactly one valid arrangement of seven queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #39?
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.