10x10 Queens Puzzle #221
Difficulty: medium. Place 10 queens on this 10x10 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 #221
What follows is a complete solve of puzzle #221, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 10 queen placements and 9 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 line confined to one region.
Techniques this board needs: last square in a region, region confined to one line, region shadow, line confined to one region, last square in a column.
Show the full walkthrough (15 steps) — contains the solution
Step 1. Only row 10, column 2 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 2, the rest of the sea green region and every square adjacent to it — diagonals included.
Step 2. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 9 squares come off.
Step 3. The slate region has 4 squares left, spread across rows 3–5, columns 6 and 7. 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.
Step 4. All of the pale yellow region's remaining squares (2 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.
Step 5. Amber is down to 4 candidates in rows 2 and 3, columns 3–5, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about amber itself.
The board after step 5 — 1 of 10 queens placed. Step 6. All of the khaki region's remaining squares (2 open squares) lie in row 8. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 11 squares come off.
Step 7. Only row 7, column 1 is still open in pale yellow, and since each colour must take a queen, that settles it. Everything else in row 7, in column 1, in pale yellow and in the ring around that square is now dead.
Step 8. The red region is down to one open square, at row 6, column 9. Every colour holds exactly one queen, so that square is forced. Row 6 and column 9 are now spoken for, along with the squares touching that queen.
Step 9. Only row 1, column 10 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 10, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 10. All of the blue region's remaining squares (3 open squares) lie in column 8. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
The board after step 10 — 4 of 10 queens placed. Step 11. Row 9 has only orange squares left (2 open squares). The row needs a queen and only orange can provide one, which pins that colour to this row and kills its squares elsewhere: 3 squares.
Step 12. Column 3 now has exactly one open square, row 2, column 3, in the amber region. Columns must each hold a queen, so that placement is forced. Everything else in row 2, in column 3, in amber and in the ring around that square is now dead.
Step 13. Blue cannot leave column 8 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 1 square in all.
Step 14. Mint is down to 4 candidates in rows 3–5, columns 4–6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint 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: slate has one square left, so its queen takes row 5, column 7; blue has one square left, so its queen takes row 3, column 8; mint has one square left, so its queen takes row 4, column 5; khaki has one square left, so its queen takes row 8, column 6; orange has one square left, so its queen takes row 9, column 4. 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 9, column 4.
The board after step 15 — 10 of 10 queens placed.
That completes the board: 10 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 10x10 Queens puzzle #221
Puzzle #221 is one of the medium-rated boards in the 10x10 set. Ten colors divide its 100 squares, and you win by placing ten queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
As with every puzzle in this collection, #221 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 #221 are: sea green (1 square), indigo (2 squares), khaki (2 squares), slate (4 squares), pale yellow (4 squares), red (9 squares), amber (12 squares), mint (13 squares), blue (18 squares), orange (35 squares). An even split would hand every color 10 squares, so the sea green region is running 9 squares under that average while the orange region is the roomiest area on the grid with 35 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 sea green, with 1 square, is under far more pressure than orange with 35. A 34-square gap like this one rewards players who start where the choices are fewest.
Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×2 box that encloses it, while orange is the most scattered at 44% of its 8×10 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
Below is every color on #221 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Sea green occupies 1 square within rows 10–10, columns 2–2. It never leaves row 10, so its queen is locked to that row.
Indigo: 2 squares across rows 1–1, columns 9–10. It never leaves row 1, so its queen is locked to that row.
Khaki covers 2 squares, spanning 1 row and 2 columns (rows 8–8, columns 5–6). It never leaves row 8, so its queen is locked to that row.
Slate occupies 4 squares within rows 3–5, columns 6–7.
Pale yellow: 4 squares across rows 7–10, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Red covers 9 squares, spanning 3 rows and 4 columns (rows 6–8, columns 7–10).
Amber occupies 12 squares within rows 1–3, columns 1–5.
Mint: 13 squares across rows 1–7, columns 4–7.
Blue covers 18 squares, spanning 7 rows and 5 columns (rows 1–7, columns 6–10).
Orange occupies 35 squares within rows 3–10, columns 1–10.
Where to start on this board
Begin where the choices are fewest. On this board that means sea green, which owns just 1 square inside a 1×1 window running from row 10 to row 10 and column 2 to column 2. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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 khaki 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 #221
A quick way to find the next move is to count how many different colors touch each line. On this board, row 9 is the narrowest, touched by only 2 colors. Column 3 is the tightest column, 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 9 rows. Row 7 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
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 100 squares of this grid. First: has any of the ten regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 10x10 boards almost always run one of those checks and forget the other.
Resist the plausible-looking queen. On #221 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 #221 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers five single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 34. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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 10x10
At 10x10 a full re-scan after every move starts to get slow, so work locally instead. Once a queen goes down, check only the regions, rows and columns that placement actually touched — that is where new information appeared. Fall back to a complete sweep only when the local pass turns up nothing, and your solve times will drop noticeably.
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 #221, start hunting for it near row 9.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 7 queens in the middle, including 1 in a corner. 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 10x10 Queens puzzle #221?
Puzzle #221 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #221 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #221, has been verified to have exactly one valid arrangement of ten queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #221?
Start with the sea green region — the most constrained color on this board, with 1 square spanning rows 10–10 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.