10x10 Queens Puzzle #240
Difficulty: hard. 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 20-step reasoning is written out below.
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Step-by-step solution for puzzle #240
What follows is a complete solve of puzzle #240, worked the way a person works it rather than the way a computer does. It runs to 20 deductions — 10 queen placements and 10 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 confined to one line, region shadow, line confined to one region, proof by contradiction, pairing argument on rows, last square in a column, last square in a row.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Only row 9, column 10 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 10, the rest of the orange region and every square adjacent to it — diagonals included.
Step 2. All of the blue 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: 9 squares come off.
Step 3. Indigo cannot leave column 2 — 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: 7 squares in all.
Step 4. Wherever mint ends up among its 3 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
The board after step 4 — 1 of 10 queens placed. Step 5. Khaki is down to 4 candidates in rows 6–8, columns 3–5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 6. Every open square left in row 10 (6 open squares) belongs to the sea green region. Since that row must contain a queen, sea green is the colour that supplies it — so sea green squares anywhere else on the board are out, 4 squares in all.
Step 7. Column 9 has only red squares left (6 open squares). The column needs a queen and only red can provide one, which pins that colour to this column and kills its squares elsewhere: 7 squares.
Step 8. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 2. Follow the forced moves that would create and column 3 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 9. Only row 1, column 2 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 2, the rest of the indigo region and every square adjacent to it — diagonals included.
The board after step 9 — 2 of 10 queens placed. Step 10. Here is the pairing argument. The amber, blue, mint, and slate regions have no open squares outside rows 2–5 — 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 11. Only row 6, column 9 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 9, in red and in the ring around that square is now dead.
Step 12. Column 8 now has exactly one open square, row 10, column 8, in the sea green region. Columns must each hold a queen, so that placement is forced. Everything else in row 10, in column 8, in sea green and in the ring around that square is now dead.
Step 13. Wherever pale yellow 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 13 — 4 of 10 queens placed. Step 14. The khaki region is down to one open square, at row 8, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 5, the remaining khaki squares, and the ring of neighbours around row 8, column 5.
Step 15. Only row 7, column 7 is still open in pale yellow, and since each colour must take a queen, that settles it. Everything else in row 7, in column 7, in pale yellow and in the ring around that square is now dead.
Step 16. All of the slate 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: 2 squares come off.
Step 17. 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 2 is down to one square, putting a queen on row 2, column 4; mint has one square left, so its queen takes row 4, column 3; blue has one square left, so its queen takes row 3, column 1; slate has one square left, so its queen takes row 5, column 6. 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 6.
The board after step 17 — 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 #240
This is puzzle number 240 in the 10x10 collection, rated hard. The grid holds 100 squares split between ten colored regions, and a finished board carries ten 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, #240 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 #240 are: orange (1 square), blue (2 squares), indigo (4 squares), mint (7 squares), pale yellow (8 squares), slate (9 squares), amber (13 squares), khaki (18 squares), red (19 squares), sea green (19 squares). An even split would hand every color 10 squares, so the orange region is running 9 squares under that average while the sea green region is the roomiest area on the grid with 19 squares.
Every color takes one queen and one only, whatever its size — which is why the 18 square difference between orange and sea green 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.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 2×2 frame; pale yellow is the loosest, occupying only 44% of its 3×6 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Below is every color on #240 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.
Orange: 1 square across rows 9–9, columns 10–10. It never leaves row 9, so its queen is locked to that row.
Blue covers 2 squares, spanning 2 rows and 1 column (rows 3–4, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.
Indigo occupies 4 squares within rows 1–2, columns 1–2.
Mint: 7 squares across rows 3–5, columns 1–4.
Pale yellow covers 8 squares, spanning 3 rows and 6 columns (rows 6–8, columns 3–8).
Slate occupies 9 squares within rows 1–5, columns 5–8.
Amber: 13 squares across rows 1–4, columns 3–6.
Khaki covers 18 squares, spanning 6 rows and 5 columns (rows 5–10, columns 1–5).
Red occupies 19 squares within rows 1–6, columns 6–10.
Sea green: 19 squares across rows 7–10, columns 3–10.
Where to start on this board
Orange is the natural opening on #240: it is the smallest region on the grid at 1 square, boxed into rows 9–9 and columns 10–10. 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 orange never leaves row 9, its queen must be in that row — and that instantly settles row 9 for everyone else. Sweep along it and X out every square that is not orange. 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 blue 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 #240
A quick way to find the next move is to count how many different colors touch each line. On this board, row 10 is the narrowest, touched by only 2 colors. Column 9 is the tightest column, 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 9 rows. Row 4 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.
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 #240 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 #240 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers three single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 18. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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 hard board from the rest.
If #240 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.
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 #240 is row 10.
A non-spoiling distribution check: solved, #240 carries 4 queens on the perimeter against 6 queens inside, with the corners empty. 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 #240?
Puzzle #240 is rated hard. 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 10x10 puzzle #240 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #240, 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 #240?
Start with the orange region — the most constrained color on this board, with 1 square spanning rows 9–9 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.