7x7 Queens Puzzle #37
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 14-step reasoning is written out below.
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Step-by-step solution for puzzle #37
What follows is a complete solve of puzzle #37, worked the way a person works it rather than the way a computer does. It runs to 14 deductions — 7 queen placements and 7 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 rows.
Techniques this board needs: last square in a region, region shadow, region confined to one line, pairing argument on rows, last square in a row.
Show the full walkthrough (9 steps) — contains the solution
Step 1. 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 2. Indigo is down to 8 candidates in rows 1–3, columns 1, 2, 3, and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. Mint cannot leave column 7 — every one of its open squares (3 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 4 squares in all.
The board after step 3 — 1 of 7 queens placed. Step 4. Wherever amber ends up among its 6 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Slate is down to 3 candidates in rows 4–6, columns 5 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.
Step 6. The red region has 4 squares left, spread across rows 4–6, columns 1 and 2. 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.
The board after step 6 — 1 of 7 queens placed. Step 7. Wherever blue ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 8. Here is the pairing argument. The mint, slate, and red regions have no open squares outside rows 4–6 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 4 squares come off in one stroke.
Step 9. 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 2; row 2 is down to one square, putting a queen on row 2, column 6; slate has one square left, so its queen takes row 5, column 5; red has one square left, so its queen takes row 6, column 1; indigo has one square left, so its queen takes row 1, column 3; mint has one square left, so its queen takes row 4, column 7. 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 7.
The board after step 9 — 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 #37
Puzzle #37 is one of the easy-rated boards in the 7x7 set. Seven colors divide its 49 squares, and you win by placing seven queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
Puzzle #37 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 #37 are: pale yellow (1 square), mint (4 squares), blue (8 squares), red (8 squares), indigo (9 squares), slate (9 squares), amber (10 squares). An even split would hand every color 7 squares, so the pale yellow region is running 6 squares under that average while the amber region is the roomiest area on the grid with 10 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So pale yellow, with 1 square, is under far more pressure than amber with 10. A 9-square gap like this one rewards players who start where the choices are fewest.
Shape matters as much as size. The mint region is the most compact color here, filling 100% of the 4×1 box that encloses it, while indigo is the most scattered at 60% of its 3×5 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.
Pale yellow covers 1 square, spanning 1 row and 1 column (rows 7–7, columns 4–4). It never leaves row 7, so its queen is locked to that row.
Mint occupies 4 squares within rows 4–7, columns 7–7. It sits wholly inside column 7, so its queen must be in that column.
Blue: 8 squares across rows 2–5, columns 2–4.
Red covers 8 squares, spanning 4 rows and 3 columns (rows 4–7, columns 1–3).
Indigo occupies 9 squares within rows 1–3, columns 1–5.
Slate: 9 squares across rows 4–7, columns 4–6.
Amber covers 10 squares, spanning 4 rows and 4 columns (rows 1–4, columns 4–7).
Where to start on this board
Pale yellow is the natural opening on #37: it is the smallest region on the grid at 1 square, boxed into rows 7–7 and columns 4–4. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.
Because pale yellow never leaves row 7, its queen must be in that row — and that instantly settles row 7 for everyone else. Sweep along it and X out every square that is not pale yellow. 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.
Nothing on this grid is pinned to one or two lines, which is why the opening takes patience. Scan the columns for one where only two or three colors appear — whichever region is forced into it claims that column outright. Chaining two of those observations is normally enough to break #37 open.
The tightest lines on puzzle #37
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Columns 1, 6 and 7 are the tightest columns, with 2 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 6 rows. Row 4, meeting 5 colors, is the opposite case — the busiest line on #37 and the last one you should expect to crack.
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.
The whole middle game reduces to two sweeps. Sweep the seven colors looking for one with a single square left; sweep the seven rows and seven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 7x7 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 #37 is rated easy
Ratings track how generous the opening is. On #37 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 9-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.
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 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.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #37, start hunting for it near row 1.
Here is a shape check that gives nothing away: when #37 is solved, 4 queens sit on the outer edge of the grid and 3 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 7x7 Queens puzzle #37?
Puzzle #37 is rated easy. 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 7x7 puzzle #37 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #37, 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 #37?
Start with the pale yellow region — the most constrained color on this board, with 1 square spanning rows 7–7 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.