10x10 Queens Puzzle #264
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 #264
What follows is a complete solve of puzzle #264, 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 region shadow.
Techniques this board needs: region shadow, region confined to one line, last square in a region.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Wherever indigo 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 2. Amber is down to 7 candidates in rows 1–3, columns 2–4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 3. Slate cannot leave column 10 — every one of its open squares (3 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 8 squares in all.
Step 4. The indigo region is down to one open square, at row 1, column 9. Every colour holds exactly one queen, so that square is forced. Row 1 and column 9 are now spoken for, along with the squares touching that queen.
Step 5. Amber is down to 4 candidates in rows 2 and 3, columns 2 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
The board after step 5 — 1 of 10 queens placed. Step 6. The khaki region has 5 squares left, spread across rows 8–10, columns 3–5. 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. Only row 10, column 5 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 5, in sea green and in the ring around that square is now dead.
Step 8. Khaki is down to 2 candidates in rows 8 and 9, columns 3 and 4, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about khaki itself.
Step 9. Orange cannot leave row 9 — every one of its open squares (2 open squares) sits there. That row belongs to orange, so every square on it owned by another colour is dead: 4 squares in all.
Step 10. The khaki region is down to one open square, at row 8, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 4, the remaining khaki squares, and the ring of neighbours around row 8, column 4.
Step 11. Amber cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 7 squares in all.
The board after step 11 — 3 of 10 queens placed. Step 12. The orange region is down to one open square, at row 9, column 1. Every colour holds exactly one queen, so that square is forced. Row 9 and column 1 are now spoken for, along with the squares touching that queen.
Step 13. Only row 4, column 3 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 3, the rest of the mint region and every square adjacent to it — diagonals included.
Step 14. The amber region is down to one open square, at row 2, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 2, the remaining amber squares, and the ring of neighbours around row 2, column 2.
Step 15. Blue cannot leave row 3 — every one of its open squares (2 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 2 squares in all.
Step 16. 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: slate has one square left, so its queen takes row 5, column 10; red has one square left, so its queen takes row 6, column 8; pale yellow has one square left, so its queen takes row 7, column 6; blue has one square left, so its queen takes row 3, column 7. 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 3, column 7.
The board after step 16 — 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 #264
This is puzzle number 264 in the 10x10 collection, rated medium. 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.
As with every puzzle in this collection, #264 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 #264 are: sea green (2 squares), indigo (3 squares), slate (3 squares), orange (4 squares), khaki (5 squares), amber (7 squares), blue (13 squares), red (20 squares), mint (21 squares), pale yellow (22 squares). An even split would hand every color 10 squares, so the sea green region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 22 squares.
Every color takes one queen and one only, whatever its size — which is why the 20 square difference between sea green and pale yellow 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. Slate is the densest region on this grid at 100% of a 3×1 frame; pale yellow is the loosest, occupying only 45% of its 7×7 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Here is the full footprint of every color on puzzle #264, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Sea green: 2 squares across rows 10–10, columns 4–5. It never leaves row 10, so its queen is locked to that row.
Indigo covers 3 squares, spanning 2 rows and 2 columns (rows 1–2, columns 9–10).
Slate occupies 3 squares within rows 3–5, columns 10–10. It sits wholly inside column 10, so its queen must be in that column.
Orange: 4 squares across rows 9–10, columns 1–2.
Khaki covers 5 squares, spanning 3 rows and 3 columns (rows 8–10, columns 3–5).
Amber occupies 7 squares within rows 1–3, columns 2–4.
Blue: 13 squares across rows 1–4, columns 5–9.
Red covers 20 squares, spanning 8 rows and 4 columns (rows 3–10, columns 7–10).
Mint occupies 21 squares within rows 1–8, columns 1–5.
Pale yellow: 22 squares across rows 4–10, columns 3–9.
Where to start on this board
Sea green is the natural opening on #264: it is the smallest region on the grid at 2 squares, boxed into rows 10–10 and columns 4–5. Since one of those squares must hold a queen, that block of 1 row and 2 columns is the most contested part of the board.
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 orange 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 #264
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 6 and 7 are the narrowest, touched by only 3 colors. Columns 1 and 6 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 3 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 10 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.
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 #264 is rated medium
The rating reflects how much the board concedes up front. Puzzle #264 has two regions confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 20 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.
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.
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 #264 is column 1.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 6 queens in the middle, and nothing in the corners. 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 #264?
Puzzle #264 is rated medium. Its smallest color region holds 2 squares, 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 #264 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #264, 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 #264?
Start with the sea green region — the most constrained color on this board, with 2 squares spanning rows 10–10 and columns 4–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.