10x10 Queens Puzzle #76
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 15-step reasoning is written out below.
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Step-by-step solution for puzzle #76
What follows is a complete solve of puzzle #76, worked the way a person works it rather than the way a computer does. It runs to 15 deductions — 10 queen placements and 5 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: last square in a region, region shadow, region confined to one line.
Show the full walkthrough (11 steps) — contains the solution
Step 1. Only row 1, column 9 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 9, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The sea green region is down to one open square, at row 10, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 1, the remaining sea green squares, and the ring of neighbours around row 10, column 1.
Step 3. Only row 9, column 3 is still open in orange, and since each colour must take a queen, that settles it. Everything else in row 9, in column 3, in orange and in the ring around that square is now dead.
Step 4. Wherever amber 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 — 3 of 10 queens placed. Step 5. Blue is down to 3 candidates in rows 3–5, columns 8 and 10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 6. The red region has 4 squares left, spread across rows 6–8, columns 7 and 8. 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 7. Blue cannot leave column 10 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 3 squares in all.
The board after step 7 — 3 of 10 queens placed. 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.
Step 9. Only row 6, column 8 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 8, the rest of the red region and every square adjacent to it — diagonals included.
Step 10. All of the amber 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 11. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: mint has one square left, so its queen takes row 4, column 5; amber has one square left, so its queen takes row 2, column 6; blue has one square left, so its queen takes row 3, column 10; pale yellow has one square left, so its queen takes row 7, column 4; slate has one square left, so its queen takes row 5, column 2. 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 2.
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 #76
This is puzzle number 76 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.
Puzzle #76 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 #76 are: indigo (1 square), sea green (1 square), amber (3 squares), red (4 squares), orange (5 squares), mint (10 squares), blue (15 squares), pale yellow (15 squares), khaki (21 squares), slate (25 squares). An even split would hand every color 10 squares, so the indigo region is running 9 squares under that average while the slate region is the roomiest area on the grid with 25 squares.
Every color takes one queen and one only, whatever its size — which is why the 24 square difference between indigo and slate 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.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and mint the most diffuse at 50% of its 5×4 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Here is the full footprint of every color on puzzle #76, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 9–9). It never leaves row 1, so its queen is locked to that row.
Sea green occupies 1 square within rows 10–10, columns 1–1. It never leaves row 10, so its queen is locked to that row.
Amber: 3 squares across rows 2–3, columns 6–7.
Red covers 4 squares, spanning 3 rows and 2 columns (rows 6–8, columns 7–8).
Orange occupies 5 squares within rows 9–10, columns 2–4.
Mint: 10 squares across rows 2–6, columns 5–8.
Blue covers 15 squares, spanning 5 rows and 6 columns (rows 1–5, columns 5–10).
Pale yellow occupies 15 squares within rows 5–9, columns 3–6.
Khaki: 21 squares across rows 5–10, columns 5–10.
Slate covers 25 squares, spanning 9 rows and 4 columns (rows 1–9, columns 1–4).
Where to start on this board
Indigo is the natural opening on #76: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 9–9. 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 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. Sea green, amber 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 #76
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 4 and 10 are the narrowest, touched by only 3 colors. Columns 1, 2 and 10 are the tightest columns, with 2 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 6, meeting 5 colors, is the opposite case — the busiest line on #76 and the last one you should expect to crack.
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.
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.
Resist the plausible-looking queen. On #76 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 #76 is rated easy
The rating reflects how much the board concedes up front. Puzzle #76 has four regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 24 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.
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.
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 #76 is column 1.
Here is a shape check that gives nothing away: when #76 is solved, 3 queens sit on the outer edge of the grid and 7 sit inside it, with 1 queen in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 #76?
Puzzle #76 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #76 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #76, 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 #76?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 9–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.