10x10 Queens Puzzle #79
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 #79
What follows is a complete solve of puzzle #79, 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 pairing argument on columns.
Techniques this board needs: region confined to one line, region shadow, pairing argument on rows, pairing argument on columns, last square in a region, last square in a column, last square in a row.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Mint cannot leave row 4 — every one of its open squares (2 open squares) sits there. That row belongs to mint, so every square on it owned by another colour is dead: 12 squares in all.
Step 2. Indigo is down to 4 candidates in rows 1–3, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. The blue region has 4 squares left, spread across rows 1–3, columns 9 and 10. 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 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 4 candidates in rows 8–10, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
Step 6. Here is the pairing argument. The indigo, amber, and blue regions have no open squares outside rows 1–3 — 3 colours confined to 3 rows. Between them they must take one row 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 and orange are now boxed into columns 1 and 2 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 8 squares at once.
Step 8. The slate region is down to one open square, at row 5, column 3. Every colour holds exactly one queen, so that square is forced. Row 5 and column 3 are now spoken for, along with the squares touching that queen.
The board after step 8 — 1 of 10 queens placed. Step 9. The khaki region has 4 squares left, spread across rows 6–8, columns 9 and 10. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.
Step 10. Here is the pairing argument. The blue and khaki regions have no open squares outside columns 9 and 10 — 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. 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: 7 squares in all.
Step 12. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: column 2 is down to one square, putting a queen on row 1, column 2; row 3 is down to one square, putting a queen on row 3, column 10; khaki has one square left, so its queen takes row 8, column 9; orange has one square left, so its queen takes row 9, column 1; red has one square left, so its queen takes row 6, column 5; mint has one square left, so its queen takes row 4, column 6; amber has one square left, so its queen takes row 2, column 4; pale yellow has one square left, so its queen takes row 7, column 7; sea green has one square left, so its queen takes row 10, 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 10, column 8.
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 #79
10x10 puzzle #79 is rated hard. Ten colored regions cover its 100 squares, and your job is to seat ten queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Puzzle #79 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #79 are: mint (2 squares), orange (4 squares), indigo (5 squares), khaki (5 squares), blue (6 squares), sea green (6 squares), slate (10 squares), amber (11 squares), red (20 squares), pale yellow (31 squares). An even split would hand every color 10 squares, so the mint region is running 8 squares under that average while the pale yellow 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. Mint is working with 2 squares while pale yellow has 31 to spare, a gap of 29. Attack the cramped end and the roomy regions tend to resolve themselves.
Square count is only half the story — how a color is arranged counts too. Mint is the tidiest shape on #79, packing 100% of its 1×2 bounding box, whereas red straggles across its 8×5 box at just 50%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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.
Mint covers 2 squares, spanning 1 row and 2 columns (rows 4–4, columns 5–6). It never leaves row 4, so its queen is locked to that row.
Orange occupies 4 squares within rows 8–10, columns 1–2.
Indigo: 5 squares across rows 1–4, columns 1–2.
Khaki covers 5 squares, spanning 4 rows and 2 columns (rows 5–8, columns 9–10).
Blue occupies 6 squares within rows 1–4, columns 9–10.
Sea green: 6 squares across rows 9–10, columns 7–10.
Slate covers 10 squares, spanning 5 rows and 3 columns (rows 4–8, columns 1–3).
Amber occupies 11 squares within rows 1–3, columns 2–6.
Red: 20 squares across rows 3–10, columns 2–6.
Pale yellow covers 31 squares, spanning 10 rows and 6 columns (rows 1–10, columns 4–9).
Where to start on this board
Open on the tightest color. Here that is mint, confined to 2 squares between rows 4 and 4 and columns 5 and 6. Its queen has nowhere else to go, which means those 1 row and 2 columns carry more constraint than anywhere else on #79.
Because mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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. Sea green 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 #79
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 5, 6, 7, 9 and 10 are the narrowest, touched by only 4 colors. Columns 7 and 8 are the tightest columns, with 2 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 4 colors, then one of those regions has to own it, and every other region is competing for the remaining 9 rows. By contrast row 4 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
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.
With ten queens to seat and ten regions competing over 100 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 10x10 boards quickly and players who stall halfway.
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 #79 is rated hard
Ratings track how generous the opening is. On #79 you get one region pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 4 colors, and a 29-square gap between the largest and smallest regions. Those four numbers largely explain the hard 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.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
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 column 7 on this board.
As a sanity check rather than a spoiler: the finished solution to #79 puts 4 queens on the outer border and 6 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 #79?
Puzzle #79 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 4 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #79 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #79, 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 #79?
Start with the mint region — the most constrained color on this board, with 2 squares spanning rows 4–4 and columns 5–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.