7x7 Queens Puzzle #21
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 10-step reasoning is written out below.
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Step-by-step solution for puzzle #21
What follows is a complete solve of puzzle #21, worked the way a person works it rather than the way a computer does. It runs to 10 deductions — 7 queen placements and 3 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 region shadow.
Techniques this board needs: last square in a region, region confined to one line, region shadow.
Show the full walkthrough (10 steps) — contains the solution
Step 1. Only row 1, column 7 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 7, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. All of the mint region's remaining squares (3 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
Step 3. The slate region has 3 squares left, spread across rows 5 and 6, columns 2 and 3. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
The board after step 3 — 1 of 7 queens placed. Step 4. The pale yellow region is down to one open square, at row 7, column 2. Every colour holds exactly one queen, so that square is forced. Row 7 and column 2 are now spoken for, along with the squares touching that queen.
Step 5. Only row 5, column 3 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 3, the rest of the slate region and every square adjacent to it — diagonals included.
Step 6. The mint region is down to one open square, at row 4, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 1, the remaining mint squares, and the ring of neighbours around row 4, column 1.
Step 7. Wherever amber ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 7 — 4 of 7 queens placed. Step 8. The blue region is down to one open square, at row 3, column 6. Every colour holds exactly one queen, so that square is forced. Row 3 and column 6 are now spoken for, along with the squares touching that queen.
Step 9. Only row 2, column 4 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 4, the rest of the amber region and every square adjacent to it — diagonals included.
Step 10. The red region is down to one open square, at row 6, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 5, the remaining red squares, and the ring of neighbours around row 6, column 5.
The board after step 10 — 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 #21
Puzzle #21 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.
There is exactly one way to finish #21, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these seven regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #21 are: indigo (1 square), mint (3 squares), pale yellow (3 squares), slate (4 squares), red (9 squares), blue (11 squares), amber (18 squares). An even split would hand every color 7 squares, so the indigo region is running 6 squares under that average while the amber region is the roomiest area on the grid with 18 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 amber with 18. A 17-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 #21, packing 100% of its 1×1 bounding box, whereas blue straggles across its 6×3 box at just 61%. 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 easy board.
Indigo: 1 square across rows 1–1, columns 7–7. It never leaves row 1, so its queen is locked to that row.
Mint covers 3 squares, spanning 3 rows and 1 column (rows 4–6, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.
Pale yellow occupies 3 squares within rows 7–7, columns 1–3. It never leaves row 7, so its queen is locked to that row.
Slate: 4 squares across rows 5–6, columns 2–3.
Red covers 9 squares, spanning 3 rows and 3 columns (rows 5–7, columns 4–6).
Blue occupies 11 squares within rows 2–7, columns 5–7.
Amber: 18 squares across rows 1–4, columns 1–6.
Where to start on this board
Indigo is the natural opening on #21: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 7–7. 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 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. Pale yellow and slate 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 #21
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2 and 3 are the narrowest, touched by only 2 colors. Columns 4 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 6, meeting 4 colors, is the opposite case — the busiest line on #21 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.
Two checks, run alternately, will carry you through the 49 squares of this grid. First: has any of the seven regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 7x7 boards almost always run one of those checks and forget the other.
Resist the plausible-looking queen. On #21 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #21 is rated easy
The easy label comes out of four measurable things about #21: four 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 17. Boards that score tight on those measures give up their first queen fast.
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.
Stuck on #21 despite the easy rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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 #21, start hunting for it near row 1.
Here is a shape check that gives nothing away: when #21 is solved, 3 queens sit on the outer edge of the grid and 4 sit inside it, with 1 queen 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 #21?
Puzzle #21 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 7x7 puzzle #21 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #21, 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 #21?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.