7x7 Queens Puzzle #79
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 15-step reasoning is written out below.
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Step-by-step solution for puzzle #79
What follows is a complete solve of puzzle #79, 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, line confined to one region, pairing argument on columns, proof by contradiction.
Show the full walkthrough (11 steps) — contains the solution
Step 1. Only row 7, column 6 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 6, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 2. The slate region is down to one open square, at row 5, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 7, the remaining slate squares, and the ring of neighbours around row 5, column 7.
Step 3. The indigo region has 4 squares left, spread across rows 1 and 2, columns 3–5. 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. Wherever blue 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.
The board after step 4 — 2 of 7 queens placed. Step 5. Mint is down to 3 candidates in rows 3 and 4, columns 4 and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.
Step 6. All of the blue 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 7. Row 6 has only red squares left (3 open squares). The row needs a queen and only red can provide one, which pins that colour to this row and kills its squares elsewhere: 2 squares.
The board after step 7 — 2 of 7 queens placed. Step 8. Here is the pairing argument. The indigo and mint regions have no open squares outside columns 4 and 5 — 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: 1 square come off in one stroke.
Step 9. Test row 1, column 5 for indigo: place a queen there, run the forced consequences through, and row 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 10. Wherever indigo ends up among its 2 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. 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: blue has one square left, so its queen takes row 3, column 3; mint has one square left, so its queen takes row 4, column 5; indigo has one square left, so its queen takes row 1, column 4; amber has one square left, so its queen takes row 2, column 1; red has one square left, so its queen takes row 6, 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 6, column 2.
The board after step 11 — 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 #79
7x7 puzzle #79 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, #79 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 #79 are: pale yellow (1 square), blue (4 squares), amber (6 squares), slate (7 squares), indigo (8 squares), mint (10 squares), red (13 squares). An even split would hand every color 7 squares, so the pale yellow region is running 6 squares under that average while the red region is the roomiest area on the grid with 13 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Pale yellow is working with 1 square while red has 13 to spare, a gap of 12. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Blue is the densest region on this grid at 100% of a 2×2 frame; red is the loosest, occupying only 65% of its 5×4 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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 hard board.
Pale yellow covers 1 square, spanning 1 row and 1 column (rows 7–7, columns 6–6). It never leaves row 7, so its queen is locked to that row.
Blue occupies 4 squares within rows 2–3, columns 3–4.
Amber: 6 squares across rows 1–4, columns 1–2.
Slate covers 7 squares, spanning 3 rows and 3 columns (rows 5–7, columns 5–7).
Indigo occupies 8 squares within rows 1–2, columns 3–7.
Mint: 10 squares across rows 3–5, columns 3–7.
Red covers 13 squares, spanning 5 rows and 4 columns (rows 3–7, columns 1–4).
Where to start on this board
The most constrained color is where a solve should begin, and on #79 that is pale yellow with 1 square. It reaches from row 7 to row 7 and from column 6 to column 6 — 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 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.
Work outward to the other flat regions next. Indigo and blue 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 #79
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 6 are the narrowest, touched by only 2 colors. Columns 1 and 2 are the tightest columns, with 2 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 colors, so one of that handful must take the row and the rest are squeezed into the other 6 rows. Row 3 sits at the far end with 4 colors crossing it; it will almost certainly be among the last to settle.
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.
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.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #79 is rated hard
The hard label comes out of four measurable things about #79: two of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 12. Boards that score tight on those measures give up their first queen fast.
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 1 on this board.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 3 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #79?
Puzzle #79 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 7x7 puzzle #79 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #79, 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 #79?
Start with the pale yellow region — the most constrained color on this board, with 1 square spanning rows 7–7 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.