10x10 Queens Puzzle #135
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 13-step reasoning is written out below.
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Step-by-step solution for puzzle #135
What follows is a complete solve of puzzle #135, worked the way a person works it rather than the way a computer does. It runs to 13 deductions — 10 queen placements and 3 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 confined to one line, region shadow, last square in a region.
Show the full walkthrough (4 steps) — contains the solution
Step 1. Indigo cannot leave column 5 — every one of its open squares (2 open squares) sits there. That column belongs to indigo, so every square in 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 6 and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
The board after step 2 — 0 of 10 queens placed. Step 3. 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: 9 squares in all.
Step 4. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: orange has one square left, so its queen takes row 9, column 6; khaki has one square left, so its queen takes row 8, column 9; pale yellow has one square left, so its queen takes row 7, column 2; sea green has one square left, so its queen takes row 10, column 3; blue has one square left, so its queen takes row 3, column 4; indigo has one square left, so its queen takes row 1, column 5; amber has one square left, so its queen takes row 2, column 7; mint has one square left, so its queen takes row 4, column 10; slate has one square left, so its queen takes row 5, column 1; red has one square left, so its queen takes row 6, 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 6, column 8.
The board after step 4 — 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 #135
10x10 puzzle #135 is rated medium. 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 #135 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 #135 are: indigo (2 squares), sea green (2 squares), orange (3 squares), amber (7 squares), khaki (7 squares), mint (12 squares), red (16 squares), pale yellow (16 squares), slate (17 squares), blue (18 squares). An even split would hand every color 10 squares, so the indigo region is running 8 squares under that average while the blue region is the roomiest area on the grid with 18 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 2 squares while blue has 18 to spare, a gap of 16. 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. Indigo is the tidiest shape on #135, packing 100% of its 2×1 bounding box, whereas blue straggles across its 7×5 box at just 51%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a medium board.
Indigo: 2 squares across rows 1–2, columns 5–5. It sits wholly inside column 5, so its queen must be in that column.
Sea green covers 2 squares, spanning 1 row and 2 columns (rows 10–10, columns 2–3). It never leaves row 10, so its queen is locked to that row.
Orange occupies 3 squares within rows 9–10, columns 6–7.
Amber: 7 squares across rows 1–3, columns 6–9.
Khaki covers 7 squares, spanning 3 rows and 3 columns (rows 8–10, columns 8–10).
Mint occupies 12 squares within rows 1–5, columns 7–10.
Red: 16 squares across rows 4–9, columns 7–10.
Pale yellow covers 16 squares, spanning 4 rows and 6 columns (rows 7–10, columns 1–6).
Slate occupies 17 squares within rows 1–7, columns 1–4.
Blue: 18 squares across rows 1–7, columns 2–6.
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 2 squares between rows 1 and 2 and columns 5 and 5. Its queen has nowhere else to go, which means those 2 rows and 1 column carry more constraint than anywhere else on #135.
Because indigo stays inside column 5, its queen belongs to that column and no other region may use it. Cross out the rest of column 5 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
Work outward to the other flat regions next. Sea green 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 #135
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 6 and 8 are the narrowest, touched by only 3 colors. Column 1 is the tightest column, with 2 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 2 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
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 #135 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 #135 is rated medium
The medium label comes out of four measurable things about #135: two of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 16. Boards that score tight on those measures give up their first queen fast.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
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 1 on this board.
A non-spoiling distribution check: solved, #135 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 #135?
Puzzle #135 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 #135 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #135, 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 #135?
Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.