10x10 Queens Puzzle #41
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 32-step reasoning is written out below.
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Step-by-step solution for puzzle #41
What follows is a complete solve of puzzle #41, worked the way a person works it rather than the way a computer does. It runs to 32 deductions — 10 queen placements and 22 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 proof by contradiction.
Techniques this board needs: region confined to one line, region shadow, pairing argument on rows, pairing argument on columns, proof by contradiction, last square in a region.
Show the full walkthrough (25 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. Amber is down to 7 candidates in rows 1, 2, 3, and 5, columns 1–3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 3. The slate region has 3 squares left, spread across rows 3 and 5, columns 7–9. 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. Here is the pairing argument. The indigo, amber, blue, and slate regions have no open squares outside rows 1, 2, 3, and 5 — 4 colours confined to 4 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 4 squares come off in one stroke.
Step 5. Blue, slate, pale yellow, and sea green are now boxed into columns 7–10 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 7 squares at once.
Step 6. The red region has 3 squares left, spread across rows 6 and 7, columns 5 and 6. 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.
The board after step 6 — 0 of 10 queens placed. Step 7. Mint and red are now boxed into columns 5 and 6 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 10 squares at once.
Step 8. Indigo is down to 5 candidates in rows 1–3, columns 2–4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 9. The khaki region has 7 squares left, spread across rows 8–10, columns 2–4. 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. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 2. Follow the forced moves that would create and column 6 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 11. Test row 3, column 4 for indigo: place a queen there, run the forced consequences through, and column 7 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 12. Now a short proof by contradiction. Suppose the amber queen went to row 1, column 1. Follow the forced moves that would create and column 6 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 13. Test row 2, column 4 for indigo: place a queen there, run the forced consequences through, and column 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 13 — 0 of 10 queens placed. Step 14. 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: 3 squares come off.
Step 15. The blue region has 6 squares left, spread across rows 2 and 3, columns 7–10. 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 16. All of the slate region's remaining squares (2 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 17. Test row 3, column 1 for amber: place a queen there, run the forced consequences through, and column 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 18. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 2. Follow the forced moves that would create and column 6 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 19. Test row 3, column 3 for amber: place a queen there, run the forced consequences through, and column 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 19 — 0 of 10 queens placed. Step 20. The amber region is down to one open square, at row 2, column 1. Every colour holds exactly one queen, so that square is forced. Row 2 and column 1 are now spoken for, along with the squares touching that queen.
Step 21. Test row 1, column 4 for indigo: place a queen there, run the forced consequences through, and column 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 22. The indigo region is down to one open square, at row 1, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 3, the remaining indigo squares, and the ring of neighbours around row 1, column 3.
Step 23. Test row 3, column 8 for blue: place a queen there, run the forced consequences through, and column 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 24. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 10. Follow the forced moves that would create and column 6 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 25. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: blue has one square left, so its queen takes row 3, column 7; mint has one square left, so its queen takes row 4, column 5; slate has one square left, so its queen takes row 5, column 8; red has one square left, so its queen takes row 6, column 6; pale yellow has one square left, so its queen takes row 7, column 10; orange has one square left, so its queen takes row 9, column 2; sea green has one square left, so its queen takes row 10, column 9; khaki has one square left, so its queen takes row 8, column 4. 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 8, column 4.
The board after step 25 — 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 #41
Puzzle #41 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, #41 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 #41 are: mint (2 squares), slate (6 squares), sea green (8 squares), pale yellow (9 squares), amber (10 squares), indigo (12 squares), blue (12 squares), red (13 squares), khaki (14 squares), orange (14 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 14 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So mint, with 2 squares, is under far more pressure than orange with 14. A 12-square gap like this one rewards players who start where the choices are fewest.
A region's outline can constrain it more than its size does. Here mint is the most solid, filling 100% of its 1×2 box, and red the most diffuse at 37% of its 7×5 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 #41, 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 4–4, columns 5–6. It never leaves row 4, so its queen is locked to that row.
Slate: 6 squares across rows 3–5, columns 7–10.
Sea green covers 8 squares, spanning 4 rows and 3 columns (rows 7–10, columns 8–10).
Pale yellow occupies 9 squares within rows 5–9, columns 8–10.
Amber: 10 squares across rows 1–5, columns 1–3.
Indigo covers 12 squares, spanning 3 rows and 6 columns (rows 1–3, columns 2–7).
Blue occupies 12 squares within rows 1–4, columns 5–10.
Red: 13 squares across rows 4–10, columns 3–7.
Khaki covers 14 squares, spanning 4 rows and 5 columns (rows 7–10, columns 2–6).
Orange occupies 14 squares within rows 5–10, columns 1–4.
Where to start on this board
Begin where the choices are fewest. On this board that means mint, which owns just 2 squares inside a 1×2 window running from row 4 to row 4 and column 5 to column 6. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
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.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: mint spans only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #41
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2 and 6 are the narrowest, touched by only 3 colors. Column 1 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 9, meeting 5 colors, is the opposite case — the busiest line on #41 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.
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.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #41 is rated hard
The rating reflects how much the board concedes up front. Puzzle #41 has one region 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 12 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
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.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #41, start hunting for it near column 1.
Here is a shape check that gives nothing away: when #41 is solved, 4 queens sit on the outer edge of the grid and 6 sit inside it, and no queen lands 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 #41?
Puzzle #41 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #41 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #41, 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 #41?
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.