10x10 Queens Puzzle #206
Difficulty: easy. 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 23-step reasoning is written out below.
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Step-by-step solution for puzzle #206
What follows is a complete solve of puzzle #206, worked the way a person works it rather than the way a computer does. It runs to 23 deductions — 10 queen placements and 13 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: region shadow, region confined to one line, last square in a region, pairing argument on columns, pairing argument on rows, proof by contradiction.
Show the full walkthrough (18 steps) — contains the solution
Step 1. Wherever mint 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 2. All of the pale yellow region's remaining squares (2 open squares) lie in column 6. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 12 squares come off.
Step 3. Khaki cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 12 squares in all.
Step 4. The sea green region is down to one open square, at row 10, column 1. Every colour holds exactly one queen, so that square is forced. Row 10 and column 1 are now spoken for, along with the squares touching that queen.
Step 5. Only row 8, column 2 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 2, the rest of the khaki region and every square adjacent to it — diagonals included.
The board after step 5 — 2 of 10 queens placed. Step 6. The pale yellow region is down to one open square, at row 7, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 6, the remaining pale yellow squares, and the ring of neighbours around row 7, column 6.
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.
Step 8. Red is down to 3 candidates in rows 5, 6, and 9, columns 9 and 10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 9. The mint region has 2 squares left, spread across rows 3 and 4, columns 9 and 10. 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 9 — 3 of 10 queens placed. Step 10. All of the orange region's remaining squares (3 open squares) lie in row 9. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 3 squares come off.
Step 11. Red is down to 2 candidates in rows 5 and 6, columns 9 and 10, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about red itself.
Step 12. Here is the pairing argument. The mint and red regions have no open squares outside columns 9 and 10 — 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: 4 squares come off in one stroke.
Step 13. Indigo cannot leave column 8 — every one of its open squares (2 open squares) sits there. That column belongs to indigo, so every square in it owned by another colour is dead: 3 squares in all.
Step 14. The orange region is down to one open square, at row 9, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 7, the remaining orange squares, and the ring of neighbours around row 9, column 7.
The board after step 14 — 4 of 10 queens placed. Step 15. Blue cannot leave column 5 — every one of its open squares (3 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 2 squares in all.
Step 16. Here is the pairing argument. The indigo, amber, blue, and mint regions have no open squares outside rows 1–4 — 4 colours confined to 4 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 4 squares come off in one stroke.
Step 17. Test row 4, column 8 for indigo: place a queen there, run the forced consequences through, and column 9 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 18. 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 8; amber has one square left, so its queen takes row 2, column 3; blue has one square left, so its queen takes row 3, column 5; mint has one square left, so its queen takes row 4, column 9; red has one square left, so its queen takes row 6, column 10; slate has one square left, so its queen takes row 5, 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 5, column 4.
The board after step 18 — 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 #206
Board #206 of the 10x10 bank carries a easy rating. Its 100 squares are shared between ten colored regions, and the solution places ten queens: one per row, one per column, one per region, none of them side by side or corner to corner.
Like every board on this site, #206 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 #206 are: pale yellow (2 squares), khaki (2 squares), mint (3 squares), sea green (3 squares), amber (7 squares), indigo (8 squares), red (8 squares), orange (12 squares), blue (15 squares), slate (40 squares). An even split would hand every color 10 squares, so the pale yellow region is running 8 squares under that average while the slate region is the roomiest area on the grid with 40 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Pale yellow has only 2 places to put its queen; slate has 40. The spread between them on this board is 38 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Square count is only half the story — how a color is arranged counts too. Pale yellow is the tidiest shape on #206, packing 100% of its 2×1 bounding box, whereas slate straggles across its 8×9 box at just 56%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Here is the full footprint of every color on puzzle #206, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Pale yellow occupies 2 squares within rows 7–8, columns 6–6. It sits wholly inside column 6, so its queen must be in that column.
Khaki: 2 squares across rows 8–9, columns 2–2. It sits wholly inside column 2, so its queen must be in that column.
Mint covers 3 squares, spanning 2 rows and 2 columns (rows 3–4, columns 9–10).
Sea green occupies 3 squares within rows 9–10, columns 1–2.
Amber: 7 squares across rows 1–2, columns 1–4.
Indigo covers 8 squares, spanning 4 rows and 3 columns (rows 1–4, columns 8–10).
Red occupies 8 squares within rows 5–9, columns 9–10.
Orange: 12 squares across rows 7–10, columns 6–10.
Blue covers 15 squares, spanning 5 rows and 5 columns (rows 1–5, columns 4–8).
Slate occupies 40 squares within rows 3–10, columns 1–9.
Where to start on this board
The most constrained color is where a solve should begin, and on #206 that is pale yellow with 2 squares. It reaches from row 7 to row 8 and from column 6 to column 6 — 2 rows 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 stays inside column 6, its queen belongs to that column and no other region may use it. Cross out the rest of column 6 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
Work outward to the other flat regions next. Amber, mint, khaki and sea green 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 #206
A quick way to find the next move is to count how many different colors touch each line. On this board, row 6 is the narrowest, touched by only 2 colors. Columns 3 and 5 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 9 rows. Row 9 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.
Keep alternating between two views of the board. In the region view you ask which of the ten 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 10x10 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 #206 is rated easy
The rating reflects how much the board concedes up front. Puzzle #206 has two regions confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 38 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
If #206 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 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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #206 the place to look for one is around row 6.
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 #206?
Puzzle #206 is rated easy. Its smallest color region holds 2 squares, 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 10x10 puzzle #206 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #206, 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 #206?
Start with the pale yellow region — the most constrained color on this board, with 2 squares spanning rows 7–8 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.