10x10 Queens Puzzle #109
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 16-step reasoning is written out below.
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Step-by-step solution for puzzle #109
What follows is a complete solve of puzzle #109, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 10 queen placements and 6 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 rows.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows.
Show the full walkthrough (8 steps) — contains the solution
Step 1. Only row 8, column 4 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 4, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 2. All of the blue 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: 11 squares come off.
Step 3. Slate cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 9 squares in all.
Step 4. Wherever red 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 — 1 of 10 queens placed. Step 5. Slate and red are now boxed into rows 5 and 6 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 7 squares at once.
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: 5 squares come off in one stroke.
Step 7. Mint cannot leave row 4 — every one of its open squares (4 open squares) sits there. That row belongs to mint, so every square on it owned by another colour is dead: 2 squares in all.
Step 8. 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: pale yellow has one square left, so its queen takes row 7, column 2; 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 9; sea green has one square left, so its queen takes row 10, column 3; indigo has one square left, so its queen takes row 1, column 5; blue has one square left, so its queen takes row 3, column 6; amber has one square left, so its queen takes row 2, column 8; mint has one square left, so its queen takes row 4, column 10; orange has one square left, so its queen takes row 9, 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 9, column 7.
The board after step 8 — 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 #109
Puzzle #109 is one of the hard-rated boards in the 10x10 set. Ten colors divide its 100 squares, and you win by placing ten queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
As with every puzzle in this collection, #109 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 #109 are: khaki (1 square), blue (2 squares), slate (2 squares), red (3 squares), amber (9 squares), sea green (9 squares), mint (12 squares), indigo (13 squares), pale yellow (21 squares), orange (28 squares). An even split would hand every color 10 squares, so the khaki region is running 9 squares under that average while the orange region is the roomiest area on the grid with 28 squares.
Every color takes one queen and one only, whatever its size — which is why the 27 square difference between khaki and orange 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 blue is the most solid, filling 100% of its 2×1 box, and mint the most diffuse at 50% of its 4×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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 hard board.
Khaki covers 1 square, spanning 1 row and 1 column (rows 8–8, columns 4–4). It never leaves row 8, so its queen is locked to that row.
Blue occupies 2 squares within rows 2–3, columns 6–6. It sits wholly inside column 6, so its queen must be in that column.
Slate: 2 squares across rows 5–6, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Red covers 3 squares, spanning 2 rows and 2 columns (rows 5–6, columns 8–9).
Amber occupies 9 squares within rows 1–3, columns 7–10.
Sea green: 9 squares across rows 8–10, columns 1–4.
Mint covers 12 squares, spanning 4 rows and 6 columns (rows 2–5, columns 5–10).
Indigo occupies 13 squares within rows 1–3, columns 1–6.
Pale yellow: 21 squares across rows 3–8, columns 1–5.
Orange covers 28 squares, spanning 5 rows and 6 columns (rows 6–10, columns 5–10).
Where to start on this board
Begin where the choices are fewest. On this board that means khaki, which owns just 1 square inside a 1×1 window running from row 8 to row 8 and column 4 to column 4. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because khaki never leaves row 8, its queen must be in that row — and that instantly settles row 8 for everyone else. Sweep along it and X out every square that is not khaki. 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, slate and red 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 #109
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 4, 7, 9 and 10 are the narrowest, touched by only 2 colors. Columns 2, 3, 7 and 10 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 3 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
The middle of #109 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.
The whole middle game reduces to two sweeps. Sweep the ten colors looking for one with a single square left; sweep the ten rows and ten columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 10x10 solve from a stalled one.
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 #109 is rated hard
The hard label comes out of four measurable things about #109: four of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 27. Boards that score tight on those measures give up their first queen fast.
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.
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 #109 is row 1.
A non-spoiling distribution check: solved, #109 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 #109?
Puzzle #109 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #109 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #109, 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 #109?
Start with the khaki region — the most constrained color on this board, with 1 square spanning rows 8–8 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.