10x10 Queens Puzzle #32
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 27-step reasoning is written out below.
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Step-by-step solution for puzzle #32
What follows is a complete solve of puzzle #32, worked the way a person works it rather than the way a computer does. It runs to 27 deductions — 10 queen placements and 17 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 shadow, pairing argument on rows, last square in a region, proof by contradiction, region confined to one line, pairing argument on columns, last square in a row.
Show the full walkthrough (22 steps) — contains the solution
Step 1. Wherever indigo ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 2. Amber is down to 5 candidates in rows 1 and 2, columns 8–10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 3. The sea green region has 4 squares left, spread across rows 8–10, columns 1 and 2. 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 and amber regions have no open squares outside rows 1 and 2 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 9 squares come off in one stroke.
Step 5. Blue and mint are now boxed into rows 3 and 4 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 9 squares at once.
Step 6. The slate region is down to one open square, at row 5, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 1, the remaining slate squares, and the ring of neighbours around row 5, column 1.
The board after step 6 — 1 of 10 queens placed. Step 7. Only row 10, column 2 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 2, in sea green and in the ring around that square is now dead.
Step 8. Pale yellow is down to 3 candidates in rows 7 and 9, columns 3–5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 9. Test row 3, column 5 for blue: 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 10. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 5. Follow the forced moves that would create and the orange region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 11. Blue cannot leave column 4 — 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 11 — 2 of 10 queens placed. Step 12. The pale yellow region has 2 squares left, spread across rows 7 and 9, columns 3 and 5. 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 13. Test row 2, column 6 for indigo: place a queen there, run the forced consequences through, and the red region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
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. 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: 1 square in all.
Step 16. Now a short proof by contradiction. Suppose the mint queen went to row 3, column 7. Follow the forced moves that would create and row 7 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 17. Amber, mint, and khaki are now boxed into columns 8–10 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 7 squares at once.
The board after step 17 — 2 of 10 queens placed. Step 18. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 6. Follow the forced moves that would create and the red region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 19. Only row 1, column 5 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 5, in indigo and in the ring around that square is now dead.
Step 20. The pale yellow region is down to one open square, at row 7, column 3. Every colour holds exactly one queen, so that square is forced. Row 7 and column 3 are now spoken for, along with the squares touching that queen.
Step 21. Red cannot leave column 7 — every one of its open squares (2 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 1 square in all.
Step 22. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: row 9 is down to one square, putting a queen on row 9, column 6; red has one square left, so its queen takes row 6, column 7; khaki has one square left, so its queen takes row 8, column 10; 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 9; blue has one square left, so its queen takes row 3, 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 3, column 4.
The board after step 22 — 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 #32
This is puzzle number 32 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.
As with every puzzle in this collection, #32 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 #32 are: indigo (3 squares), sea green (4 squares), amber (6 squares), slate (7 squares), khaki (7 squares), mint (9 squares), blue (11 squares), pale yellow (15 squares), red (19 squares), orange (19 squares). An even split would hand every color 10 squares, so the indigo region is running 7 squares under that average while the orange region is the roomiest area on the grid with 19 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 3 squares while orange has 19 to spare, a gap of 16. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Khaki is the densest region on this grid at 88% of a 4×2 frame; pale yellow is the loosest, occupying only 38% of its 8×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Indigo occupies 3 squares within rows 1–2, columns 5–6.
Sea green: 4 squares across rows 8–10, columns 1–2.
Amber covers 6 squares, spanning 2 rows and 4 columns (rows 1–2, columns 7–10).
Slate occupies 7 squares within rows 1–5, columns 1–2.
Khaki: 7 squares across rows 5–8, columns 9–10.
Mint covers 9 squares, spanning 3 rows and 4 columns (rows 2–4, columns 7–10).
Blue occupies 11 squares within rows 1–4, columns 2–5.
Pale yellow: 15 squares across rows 3–10, columns 1–5.
Red covers 19 squares, spanning 6 rows and 6 columns (rows 3–8, columns 3–8).
Orange occupies 19 squares within rows 6–10, columns 4–10.
Where to start on this board
Indigo is the natural opening on #32: it is the smallest region on the grid at 3 squares, boxed into rows 1–2 and columns 5–6. Since one of those squares must hold a queen, that block of 2 rows and 2 columns is the most contested part of the board.
Indigo does not collapse onto a single line, so it will not hand you a queen on move one. What it does give you is a short list of 3 candidates. Take each one in turn, imagine the queen there, cross out its row, its column and the eight squares around it, and ask whether any other region is left with nowhere to go. One or two candidates usually fail that test immediately.
Work outward to the other flat regions next. Amber is no more than two rows tall, so its queen is 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 #32
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 9 and 10 are the narrowest, touched by only 3 colors. Columns 1, 3 and 6 are the tightest columns, with 3 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 8 is the crowded extreme at 5 colors — worth revisiting later, not now.
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.
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 #32 is rated hard
Ratings track how generous the opening is. On #32 you get no region pinned to a single row or column, a smallest color of 3 squares, a tightest row crossed by 3 colors, and a 16-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
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.
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 row 9 on this board.
A non-spoiling distribution check: solved, #32 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 #32?
Puzzle #32 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 3 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 10x10 puzzle #32 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #32, 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 #32?
Start with the indigo region — the most constrained color on this board, with 3 squares spanning rows 1–2 and columns 5–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.