7x7 Queens Puzzle #15
Difficulty: hard. 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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #15
What follows is a complete solve of puzzle #15, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 7 queen placements and 12 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: region shadow, pairing argument on columns, region confined to one line, proof by contradiction, last square in a region, last square in a column.
Show the full walkthrough (15 steps) — contains the solution
Step 1. Wherever amber 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 2. Mint is down to 3 candidates in rows 4 and 5, columns 4 and 5, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about mint itself.
Step 3. The red region has 4 squares left, spread across rows 4–6, columns 5–7. 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. Here is the pairing argument. The blue, red, and pale yellow regions have no open squares outside columns 5–7 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 4 squares come off in one stroke.
Step 5. Indigo is down to 5 candidates in rows 1, 2, 3, and 5, columns 3 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
The board after step 5 — 0 of 7 queens placed. Step 6. The amber region has 3 squares left, spread across rows 1 and 2, 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.
Step 7. Mint cannot leave column 4 — every one of its open squares (2 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 5 squares in all.
Step 8. All of the indigo 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: 3 squares come off.
Step 9. Amber cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 4 squares in all.
Step 10. Now a short proof by contradiction. Suppose the indigo queen went to row 3, column 3. Follow the forced moves that would create and column 5 runs out of open squares entirely — impossible. So that square is ruled out for good.
The board after step 10 — 0 of 7 queens placed. Step 11. Only row 1, column 3 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 3, in indigo and in the ring around that square is now dead.
Step 12. The amber region is down to one open square, at row 2, column 1. Every colour holds exactly one queen, so that square is forced. Row 2 and column 1 are now spoken for, along with the squares touching that queen.
Step 13. 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 14. Red is down to 3 candidates in rows 5 and 6, columns 5–7, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about red itself.
Step 15. 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; column 5 is down to one square, putting a queen on row 6, column 5; pale yellow has one square left, so its queen takes row 7, column 7; blue has one square left, so its queen takes row 3, column 6; slate has one square left, so its queen takes row 5, column 2. 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 5, column 2.
The board after step 15 — 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 #15
7x7 puzzle #15 is rated hard. 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.
As with every puzzle in this collection, #15 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.
How the colors divide this board
From smallest to largest, the regions on puzzle #15 are: mint (3 squares), amber (4 squares), pale yellow (4 squares), red (5 squares), blue (8 squares), indigo (10 squares), slate (15 squares). An even split would hand every color 7 squares, so the mint region is running 4 squares under that average while the slate region is the roomiest area on the grid with 15 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Mint is working with 3 squares while slate has 15 to spare, a gap of 12. 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 pale yellow is the most solid, filling 100% of its 2×2 box, and indigo the most diffuse at 50% of its 5×4 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 #15, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Mint: 3 squares across rows 4–5, columns 4–5.
Amber covers 4 squares, spanning 2 rows and 3 columns (rows 1–2, columns 1–3).
Pale yellow occupies 4 squares within rows 6–7, columns 6–7.
Red: 5 squares across rows 4–6, columns 5–7.
Blue covers 8 squares, spanning 4 rows and 3 columns (rows 1–4, columns 5–7).
Indigo occupies 10 squares within rows 1–5, columns 2–5.
Slate: 15 squares across rows 3–7, columns 1–5.
Where to start on this board
The most constrained color is where a solve should begin, and on #15 that is mint with 3 squares. It reaches from row 4 to row 5 and from column 4 to column 5 — 2 rows by 2 columns. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
Mint spans 2 rows, so it stays open for now — but it is still the shortest list on the board at 3 squares. Work through those squares first and mark the ones that would leave another region homeless. That pruning is what makes the rest of the grid legible.
Work outward to the other flat regions next. 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 #15
A quick way to find the next move is to count how many different colors touch each line. On this board, row 7 is the narrowest, touched by only 2 colors. Column 1 is the tightest column, with 2 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 6 rows. By contrast row 4 meets 5 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
Working through the middle game
The middle of #15 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.
With seven queens to seat and seven regions competing over 49 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 7x7 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 #15 is rated hard
The rating reflects how much the board concedes up front. Puzzle #15 has no region confined to a single row or column, its smallest color holds 3 squares, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 12 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.
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 7 on this board.
As a sanity check rather than a spoiler: the finished solution to #15 puts 3 queens on the outer border and 4 queens in the interior, 1 of them in an actual corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 #15?
Puzzle #15 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 2 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 7x7 puzzle #15 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #15, 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 #15?
Start with the mint region — the most constrained color on this board, with 3 squares spanning rows 4–5 and columns 4–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.