10x10 Queens Puzzle #44
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 #44
What follows is a complete solve of puzzle #44, 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 region shadow.
Techniques this board needs: region confined to one line, last square in a region, region shadow.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Mint cannot leave column 4 — every one of its open squares (2 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 12 squares in all.
Step 2. The slate region is down to one open square, at row 5, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 6, the remaining slate squares, and the ring of neighbours around row 5, column 6.
Step 3. Only row 4, column 4 is still open in mint, and since each colour must take a queen, that settles it. Everything else in row 4, in column 4, in mint and in the ring around that square is now dead.
Step 4. Wherever indigo 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 — 2 of 10 queens placed. Step 5. Only row 3, column 2 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 2, the rest of the blue region and every square adjacent to it — diagonals included.
Step 6. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.
Step 7. Amber cannot leave row 2 — every one of its open squares (2 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 3 squares in all.
Step 8. All of the red region's remaining squares (2 open squares) lie in row 6. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.
The board after step 8 — 3 of 10 queens placed. Step 9. Only row 7, column 3 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 3, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 10. The indigo region is down to one open square, at row 1, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 1, the remaining indigo squares, and the ring of neighbours around row 1, column 1.
Step 11. Khaki is down to 5 candidates in rows 8–10, columns 8–10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 12. 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: sea green has one square left, so its queen takes row 10, column 9; red has one square left, so its queen takes row 6, column 10; khaki has one square left, so its queen takes row 8, column 8; orange has one square left, so its queen takes row 9, column 5; amber has one square left, so its queen takes row 2, 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 2, column 7.
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 #44
This is puzzle number 44 in the 10x10 collection, rated hard. 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.
There is exactly one way to finish #44, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these ten regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #44 are: mint (2 squares), slate (2 squares), sea green (2 squares), indigo (6 squares), blue (7 squares), khaki (10 squares), pale yellow (12 squares), amber (17 squares), red (17 squares), orange (25 squares). An even split would hand every color 10 squares, so the mint region is running 8 squares under that average while the orange 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 23 square difference between mint 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 mint is the most solid, filling 100% of its 2×1 box, and indigo the most diffuse at 50% of its 3×4 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 #44, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Mint occupies 2 squares within rows 4–5, columns 4–4. It sits wholly inside column 4, so its queen must be in that column.
Slate: 2 squares across rows 5–5, columns 5–6. It never leaves row 5, so its queen is locked to that row.
Sea green covers 2 squares, spanning 2 rows and 1 column (rows 9–10, columns 9–9). It sits wholly inside column 9, so its queen must be in that column.
Indigo occupies 6 squares within rows 1–3, columns 1–4.
Blue: 7 squares across rows 2–5, columns 1–3.
Khaki covers 10 squares, spanning 6 rows and 3 columns (rows 5–10, columns 8–10).
Pale yellow occupies 12 squares within rows 5–8, columns 1–5.
Amber: 17 squares across rows 1–4, columns 3–9.
Red covers 17 squares, spanning 6 rows and 5 columns (rows 1–6, columns 6–10).
Orange occupies 25 squares within rows 7–10, columns 1–8.
Where to start on this board
Mint is the natural opening on #44: it is the smallest region on the grid at 2 squares, boxed into rows 4–5 and columns 4–4. Since one of those squares must hold a queen, that block of 2 rows and 1 column is the most contested part of the board.
Because mint stays inside column 4, its queen belongs to that column and no other region may use it. Cross out the rest of column 4 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. Slate 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 #44
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 6, 7, 8, 9 and 10 are the narrowest, touched by only 3 colors. Column 10 is the tightest column, 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 5, meeting 6 colors, is the opposite case — the busiest line on #44 and the last one you should expect to crack.
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.
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 #44 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 #44 is rated hard
The rating reflects how much the board concedes up front. Puzzle #44 has three regions confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 23 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
Stuck on #44 despite the hard rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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 #44 is column 10.
Here is a shape check that gives nothing away: when #44 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 #44?
Puzzle #44 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #44 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #44, 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 #44?
Start with the mint region — the most constrained color on this board, with 2 squares spanning rows 4–5 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.