10x10 Queens Puzzle #85
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 17-step reasoning is written out below.
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Step-by-step solution for puzzle #85
What follows is a complete solve of puzzle #85, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 10 queen placements and 7 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 shadow, region confined to one line, last square in a row.
Show the full walkthrough (11 steps) — contains the solution
Step 1. Only row 10, column 6 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 6, the rest of the sea green region and every square adjacent to it — diagonals included.
Step 2. The orange region is down to one open square, at row 9, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 10, the remaining orange squares, and the ring of neighbours around row 9, column 10.
Step 3. The indigo region has 6 squares left, spread across rows 1–3, columns 1–3. 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 amber ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 4 — 2 of 10 queens placed. Step 5. Red is down to 4 candidates in rows 6 and 7, columns 4, 5, and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 6. The pale yellow region has 3 squares left, spread across rows 7 and 8, columns 7 and 8. 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. Khaki cannot leave row 8 — every one of its open squares (2 open squares) sits there. That row belongs to khaki, so every square on it owned by another colour is dead: 7 squares in all.
The board after step 7 — 2 of 10 queens placed. Step 8. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: pale yellow has one square left, so its queen takes row 7, column 7; red has one square left, so its queen takes row 6, column 5; khaki has one square left, so its queen takes row 8, column 4; amber has one square left, so its queen takes row 2, column 3. 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 2, column 3.
Step 9. All of the indigo 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: 2 squares come off.
Step 10. 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 11. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: row 4 is down to one square, putting a queen on row 4, column 2; indigo has one square left, so its queen takes row 1, column 1; blue has one square left, so its queen takes row 3, column 9; slate has one square left, so its queen takes row 5, column 8. 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 8.
The board after step 11 — 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 #85
Puzzle #85 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.
There is exactly one way to finish #85, 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 ten regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #85 are: sea green (1 square), orange (3 squares), indigo (6 squares), amber (6 squares), pale yellow (9 squares), blue (11 squares), red (11 squares), khaki (12 squares), slate (13 squares), mint (28 squares). An even split would hand every color 10 squares, so the sea green region is running 9 squares under that average while the mint region is the roomiest area on the grid with 28 squares.
Every color takes one queen and one only, whatever its size — which is why the 27 square difference between sea green and mint 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 sea green region is the most compact color here, filling 100% of the 1×1 box that encloses it, while red is the most scattered at 46% of its 6×4 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 #85 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 covers 1 square, spanning 1 row and 1 column (rows 10–10, columns 6–6). It never leaves row 10, so its queen is locked to that row.
Orange occupies 3 squares within rows 9–10, columns 9–10.
Indigo: 6 squares across rows 1–3, columns 1–3.
Amber covers 6 squares, spanning 3 rows and 4 columns (rows 1–3, columns 2–5).
Pale yellow occupies 9 squares within rows 7–10, columns 7–10.
Blue: 11 squares across rows 1–4, columns 8–10.
Red covers 11 squares, spanning 6 rows and 4 columns (rows 4–9, columns 4–7).
Khaki occupies 12 squares within rows 8–10, columns 1–5.
Slate: 13 squares across rows 3–7, columns 7–10.
Mint covers 28 squares, spanning 8 rows and 7 columns (rows 1–8, columns 1–7).
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 6 to column 6. 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. Orange is no more than two rows tall, so its queen is 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 #85
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 5 and 6 are the narrowest, touched by only 3 colors. Columns 1, 6 and 8 are the tightest columns, with 3 colors.
A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 9 rows. Row 3 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.
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.
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 #85 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 27. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
Stuck on #85 despite the medium 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 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.
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 #85 is row 5.
A non-spoiling distribution check: solved, #85 carries 3 queens on the perimeter against 7 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 10x10 Queens puzzle #85?
Puzzle #85 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #85 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #85, 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 #85?
Start with the sea green region — the most constrained color on this board, with 1 square spanning rows 10–10 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.