10x10 Queens Puzzle #8
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 17-step reasoning is written out below.
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Step-by-step solution for puzzle #8
What follows is a complete solve of puzzle #8, 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 pairing argument on columns.
Techniques this board needs: region confined to one line, region shadow, pairing argument on rows, last square in a region, pairing argument on columns, last square in a row.
Show the full walkthrough (11 steps) — contains the solution
Step 1. 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: 12 squares in all.
Step 2. Orange is down to 3 candidates in rows 9 and 10, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
Step 3. Indigo and amber are now boxed into rows 1 and 2 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 6 squares at once.
Step 4. The mint region is down to one open square, at row 4, column 6. Every colour holds exactly one queen, so that square is forced. Row 4 and column 6 are now spoken for, along with the squares touching that queen.
The board after step 4 — 1 of 10 queens placed. Step 5. Only row 3, column 4 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 4, the rest of the blue region and every square adjacent to it — diagonals included.
Step 6. Here is the pairing argument. The red and orange regions have no open squares outside columns 1 and 2 — 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: 6 squares come off in one stroke.
Step 7. Look along row 2: one square survives, row 2, column 10, which belongs to amber. A row cannot be left empty, so the queen goes there. That clears row 2, column 10, the rest of the amber region and every square adjacent to it — diagonals included.
The board after step 7 — 3 of 10 queens placed. Step 8. Khaki is down to 4 candidates in rows 8 and 9, columns 7–9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 9. Khaki, orange, and sea green are now boxed into rows 8–10 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 5 squares at once.
Step 10. Here is the pairing argument. The indigo and pale yellow regions have no open squares outside columns 3 and 5 — 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 11. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: row 8 is down to one square, putting a queen on row 8, column 8; row 9 is down to one square, putting a queen on row 9, column 1; red has one square left, so its queen takes row 6, column 2; slate has one square left, so its queen takes row 5, column 9; pale yellow has one square left, so its queen takes row 7, column 5; indigo has one square left, so its queen takes row 1, column 3; sea green has one square left, so its queen takes row 10, 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 10, column 7.
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 #8
This is puzzle number 8 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.
There is exactly one way to finish #8, 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 #8 are: blue (2 squares), orange (3 squares), amber (8 squares), mint (8 squares), khaki (8 squares), indigo (10 squares), sea green (12 squares), red (13 squares), pale yellow (17 squares), slate (19 squares). An even split would hand every color 10 squares, so the blue region is running 8 squares under that average while the slate region is the roomiest area on the grid with 19 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Blue is working with 2 squares while slate has 19 to spare, a gap of 17. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Blue is the densest region on this grid at 100% of a 1×2 frame; mint is the loosest, occupying only 40% of its 4×5 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 #8 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.
Blue occupies 2 squares within rows 3–3, columns 4–5. It never leaves row 3, so its queen is locked to that row.
Orange: 3 squares across rows 9–10, columns 1–2.
Amber covers 8 squares, spanning 3 rows and 4 columns (rows 1–3, columns 7–10).
Mint occupies 8 squares within rows 1–4, columns 5–9.
Khaki: 8 squares across rows 7–10, columns 7–10.
Indigo covers 10 squares, spanning 3 rows and 5 columns (rows 1–3, columns 1–5).
Sea green occupies 12 squares within rows 8–10, columns 3–9.
Red: 13 squares across rows 2–8, columns 1–4.
Pale yellow covers 17 squares, spanning 5 rows and 6 columns (rows 5–9, columns 2–7).
Slate occupies 19 squares within rows 3–8, columns 5–10.
Where to start on this board
Blue is the natural opening on #8: it is the smallest region on the grid at 2 squares, boxed into rows 3–3 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 blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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 #8
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 4, 5, 6 and 10 are the narrowest, touched by only 3 colors. Columns 1 and 10 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 6 colors — worth revisiting later, not now.
Working through the middle game
The middle of #8 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.
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 #8 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers one single-line region, a 2-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 17. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
Stuck on #8 despite the hard 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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 1 on this board.
A non-spoiling distribution check: solved, #8 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 #8?
Puzzle #8 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #8 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #8, 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 #8?
Start with the blue region — the most constrained color on this board, with 2 squares spanning rows 3–3 and columns 4–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.