7x7 Queens Puzzle #78
Difficulty: medium. 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 13-step reasoning is written out below.
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Step-by-step solution for puzzle #78
What follows is a complete solve of puzzle #78, worked the way a person works it rather than the way a computer does. It runs to 13 deductions — 7 queen placements and 6 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, line confined to one region, pairing argument on columns, proof by contradiction, region confined to one line, last square in a row.
Show the full walkthrough (9 steps) — contains the solution
Step 1. Only row 1, column 6 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 6, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The amber region is down to one open square, at row 2, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 3, the remaining amber squares, and the ring of neighbours around row 2, column 3.
Step 3. The slate region has 3 squares left, spread across rows 4–6, columns 5 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.
The board after step 3 — 2 of 7 queens placed. Step 4. Wherever red ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 5. Row 7 has only pale yellow squares left (5 open squares). The row needs a queen and only pale yellow can provide one, which pins that colour to this row and kills its squares elsewhere: 1 square.
Step 6. Here is the pairing argument. The blue and slate regions have no open squares outside columns 5 and 7 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 2 squares come off in one stroke.
The board after step 6 — 2 of 7 queens placed. Step 7. Test row 3, column 5 for blue: place a queen there, run the forced consequences through, and the red region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 8. All of the blue region's remaining squares (3 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
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 1; pale yellow has one square left, so its queen takes row 7, column 4; mint has one square left, so its queen takes row 4, column 2; slate has one square left, so its queen takes row 5, column 5; blue has one square left, so its queen takes row 3, 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 3, 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 #78
Board #78 of the 7x7 bank carries a medium rating. Its 49 squares are shared between seven colored regions, and the solution places seven queens: one per row, one per column, one per region, none of them side by side or corner to corner.
As with every puzzle in this collection, #78 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 #78 are: indigo (1 square), red (3 squares), slate (6 squares), amber (7 squares), blue (10 squares), mint (11 squares), pale yellow (11 squares). An even split would hand every color 7 squares, so the indigo region is running 6 squares under that average while the pale yellow region is the roomiest area on the grid with 11 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 indigo, with 1 square, is under far more pressure than pale yellow with 11. A 10-square gap like this one rewards players who start where the choices are fewest.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #78, packing 100% of its 1×1 bounding box, whereas blue straggles across its 5×4 box at just 50%. 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 medium board.
Indigo: 1 square across rows 1–1, columns 6–6. It never leaves row 1, so its queen is locked to that row.
Red covers 3 squares, spanning 2 rows and 2 columns (rows 5–6, columns 1–2).
Slate occupies 6 squares within rows 4–6, columns 5–7.
Amber: 7 squares across rows 1–2, columns 1–5.
Blue covers 10 squares, spanning 5 rows and 4 columns (rows 1–5, columns 4–7).
Mint occupies 11 squares within rows 2–5, columns 1–4.
Pale yellow: 11 squares across rows 6–7, columns 1–7.
Where to start on this board
Begin where the choices are fewest. On this board that means indigo, which owns just 1 square inside a 1×1 window running from row 1 to row 1 and column 6 to column 6. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
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. Amber, red 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 #78
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 1 color. Columns 3 and 7 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 1 color has to be claimed by one of them, which pushes every remaining region into the other 6 rows. Row 5, meeting 4 colors, is the opposite case — the busiest line on #78 and the last one you should expect to crack.
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.
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 #78 is rated medium
The medium label comes out of four measurable things about #78: two 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 10. Boards that score tight on those measures give up their first queen fast.
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.
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.
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 #78, start hunting for it near row 7.
Here is a shape check that gives nothing away: when #78 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 #78?
Puzzle #78 is rated medium. 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 #78 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #78, 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 #78?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.