10x10 Queens Puzzle #80
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 18-step reasoning is written out below.
Previous puzzle · All 10x10 puzzles · Next puzzle
Step-by-step solution for puzzle #80
What follows is a complete solve of puzzle #80, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 10 queen placements and 8 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 columns, last square in a region.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Indigo cannot leave row 1 — every one of its open squares (3 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 8 squares in all.
Step 2. Blue is down to 3 candidates in rows 3 and 4, columns 1 and 2, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about blue itself.
Step 3. The slate region has 5 squares left, spread across rows 5–8, 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.
Step 4. Wherever khaki 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 — 0 of 10 queens placed. Step 5. Orange is down to 3 candidates in rows 8 and 9, columns 1 and 2, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about orange itself.
Step 6. Here is the pairing argument. The blue 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: 8 squares come off in one stroke.
Step 7. Indigo, amber, and slate are now boxed into columns 8–10 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 10 squares at once.
Step 8. The khaki region is down to one open square, at row 8, column 7. Every colour holds exactly one queen, so that square is forced. Row 8 and column 7 are now spoken for, along with the squares touching that queen.
The board after step 8 — 1 of 10 queens placed. Step 9. Only row 9, column 1 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 1, the rest of the orange region and every square adjacent to it — diagonals included.
Step 10. The blue region is down to one open square, at row 3, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 2, the remaining blue squares, and the ring of neighbours around row 3, column 2.
Step 11. Sea green cannot leave row 10 — every one of its open squares (2 open squares) sits there. That row belongs to sea green, so every square on it owned by another colour is dead: 2 squares in all.
Step 12. 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: pale yellow has one square left, so its queen takes row 7, column 3; red has one square left, so its queen takes row 6, column 5; slate has one square left, so its queen takes row 5, column 9; amber has one square left, so its queen takes row 2, column 10; indigo has one square left, so its queen takes row 1, column 8; sea green has one square left, so its queen takes row 10, column 6; mint has one square left, so its queen takes row 4, 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 4, column 4.
The board after step 12 — 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 #80
This is puzzle number 80 in the 10x10 collection, rated easy. 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, #80 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 #80 are: indigo (3 squares), blue (3 squares), orange (3 squares), slate (5 squares), khaki (6 squares), amber (10 squares), pale yellow (12 squares), sea green (12 squares), red (15 squares), mint (31 squares). An even split would hand every color 10 squares, so the indigo region is running 7 squares under that average while the mint region is the roomiest area on the grid with 31 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 3 squares while mint has 31 to spare, a gap of 28. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×3 frame; red is the loosest, occupying only 36% of its 6×7 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Indigo occupies 3 squares within rows 1–1, columns 8–10. It never leaves row 1, so its queen is locked to that row.
Blue: 3 squares across rows 3–4, columns 1–2.
Orange covers 3 squares, spanning 2 rows and 2 columns (rows 8–9, columns 1–2).
Slate occupies 5 squares within rows 5–8, columns 9–10.
Khaki: 6 squares across rows 5–8, columns 7–9.
Amber covers 10 squares, spanning 5 rows and 3 columns (rows 2–6, columns 8–10).
Pale yellow occupies 12 squares within rows 7–10, columns 1–5.
Sea green: 12 squares across rows 9–10, columns 4–10.
Red covers 15 squares, spanning 6 rows and 7 columns (rows 4–9, columns 1–7).
Mint occupies 31 squares within rows 1–6, columns 1–8.
Where to start on this board
Indigo is the natural opening on #80: it is the smallest region on the grid at 3 squares, boxed into rows 1–1 and columns 8–10. Since one of those squares must hold a queen, that block of 1 row and 3 columns is the most contested part of the board.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Blue, orange 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 #80
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2 and 10 are the narrowest, touched by only 2 colors. Columns 3 and 6 are the tightest columns, with 3 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 8 is the crowded extreme at 5 colors — worth revisiting later, not now.
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.
Resist the plausible-looking queen. On #80 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 #80 is rated easy
Ratings track how generous the opening is. On #80 you get one region pinned to a single row or column, a smallest color of 3 squares, a tightest row crossed by 2 colors, and a 28-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.
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.
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, #80 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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 10x10 Queens puzzle #80?
Puzzle #80 is rated easy. Its smallest color region holds 3 squares, its tightest row is touched by 2 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #80 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #80, 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 #80?
Start with the indigo region — the most constrained color on this board, with 3 squares spanning rows 1–1 and columns 8–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.