7x7 Queens Puzzle #16
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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #16
What follows is a complete solve of puzzle #16, 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: last square in a region, region shadow, region confined to one line, line confined to one region, proof by contradiction.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Only row 7, column 7 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 7, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 2. Amber is down to 3 candidates in rows 2 and 3, columns 5 and 6, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about amber itself.
Step 3. Mint cannot leave column 1 — 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.
Step 4. Wherever slate 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. Red is down to 5 candidates in rows 4–6, columns 4–6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
The board after step 5 — 1 of 7 queens placed. Step 6. The blue region has 4 squares left, spread across rows 3–5, columns 3 and 4. 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. Wherever slate ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 8. Every open square left in row 1 (4 open squares) belongs to the indigo region. Since that row must contain a queen, indigo is the colour that supplies it — so indigo squares anywhere else on the board are out, 3 squares in all.
Step 9. Row 2 has only amber squares left (2 open squares). The row needs a queen and only amber can provide one, which pins that colour to this row and kills its squares elsewhere: 1 square.
The board after step 9 — 1 of 7 queens placed. Step 10. All of the amber region's remaining squares (2 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 1 square come off.
Step 11. Row 3 has only blue squares left (2 open squares). The row needs a queen and only blue can provide one, which pins that colour to this row and kills its squares elsewhere: 2 squares.
Step 12. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 3. Follow the forced moves that would create and row 5 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 13. Test row 1, column 4 for indigo: place a queen there, run the forced consequences through, and row 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 14. 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: indigo has one square left, so its queen takes row 1, column 2; slate has one square left, so its queen takes row 5, column 3; blue has one square left, so its queen takes row 3, column 4; amber has one square left, so its queen takes row 2, column 6; red has one square left, so its queen takes row 6, column 5; mint has one square left, so its queen takes row 4, column 1. 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 1.
The board after step 14 — 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 #16
This is puzzle number 16 in the 7x7 collection, rated medium. The grid holds 49 squares split between seven colored regions, and a finished board carries seven queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
Like every board on this site, #16 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 #16 are: pale yellow (1 square), mint (4 squares), amber (5 squares), blue (6 squares), slate (6 squares), indigo (12 squares), red (15 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 15 squares.
Every color takes one queen and one only, whatever its size — which is why the 14 square difference between pale yellow and red is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
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 blue is the most scattered at 50% of its 4×3 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 7–7). It never leaves row 7, so its queen is locked to that row.
Mint occupies 4 squares within rows 4–7, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Amber: 5 squares across rows 1–3, columns 5–7.
Blue covers 6 squares, spanning 4 rows and 3 columns (rows 2–5, columns 3–5).
Slate occupies 6 squares within rows 4–7, columns 2–3.
Indigo: 12 squares across rows 1–4, columns 1–6.
Red covers 15 squares, spanning 5 rows and 5 columns (rows 3–7, columns 3–7).
Where to start on this board
Open on the tightest color. Here that is pale yellow, confined to 1 square between rows 7 and 7 and columns 7 and 7. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #16.
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.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: slate spans only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #16
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 and 2 are the tightest columns, with 2 colors.
A line with few colors is a line close to resolving. Where only 2 colors 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.
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 #16 is rated medium
Ratings track how generous the opening is. On #16 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 14-square gap between the largest and smallest regions. Those four numbers largely explain the medium 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 medium board from the rest.
If #16 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.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #16 is row 1.
A non-spoiling distribution check: solved, #16 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 #16?
Puzzle #16 is rated medium. 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 #16 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #16, 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 #16?
Start with the pale yellow region — the most constrained color on this board, with 1 square spanning rows 7–7 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.