10x10 Queens Puzzle #33
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 21-step reasoning is written out below.
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Step-by-step solution for puzzle #33
What follows is a complete solve of puzzle #33, worked the way a person works it rather than the way a computer does. It runs to 21 deductions — 10 queen placements and 11 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: last square in a region, region shadow, pairing argument on rows, region confined to one line, pairing argument on columns, proof by contradiction.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Only row 4, column 10 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 10, the rest of the mint region and every square adjacent to it — diagonals included.
Step 2. Indigo is down to 4 candidates in rows 1 and 2, columns 7–9, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. The amber region has 4 squares left, spread across rows 1 and 2, columns 3–5. 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.
Step 4. Wherever sea green 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 — 1 of 10 queens placed. Step 5. Indigo and amber are now boxed into rows 1 and 2 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 6 squares at once.
Step 6. All of the blue region's remaining squares (4 open squares) lie in row 3. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.
Step 7. Only row 5, column 7 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 7, in slate and in the ring around that square is now dead.
Step 8. 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: 2 squares come off.
Step 9. 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: 2 squares in all.
The board after step 9 — 2 of 10 queens placed. Step 10. Here is the pairing argument. The blue and khaki regions have no open squares outside columns 1 and 2 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 4 squares come off in one stroke.
Step 11. Only row 10, column 3 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 3, in sea green and in the ring around that square is now dead.
Step 12. The amber region is down to one open square, at row 2, column 4. Every colour holds exactly one queen, so that square is forced. Row 2 and column 4 are now spoken for, along with the squares touching that queen.
Step 13. Only row 6, column 5 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 5, the rest of the red region and every square adjacent to it — diagonals included.
The board after step 13 — 5 of 10 queens placed. Step 14. Pale yellow is down to 4 candidates in rows 7–9, columns 6, 8, and 9, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 15. The orange region has 2 squares left, spread across rows 7 and 9, columns 6 and 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 16. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 8. Follow the forced moves that would create and row 9 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 17. 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: indigo has one square left, so its queen takes row 1, column 9; orange has one square left, so its queen takes row 9, column 6; pale yellow has one square left, so its queen takes row 7, column 8; khaki has one square left, so its queen takes row 8, column 2; blue has one square left, so its queen takes row 3, column 1. 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 1.
The board after step 17 — 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 #33
Puzzle #33 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.
There is exactly one way to finish #33, 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 #33 are: mint (1 square), amber (4 squares), indigo (5 squares), sea green (5 squares), khaki (8 squares), slate (9 squares), blue (12 squares), orange (12 squares), red (16 squares), pale yellow (28 squares). An even split would hand every color 10 squares, so the mint region is running 9 squares under that average while the pale yellow 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 mint and pale yellow 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 1×1 box, and sea green the most diffuse at 56% of its 3×3 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 #33, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Mint: 1 square across rows 4–4, columns 10–10. It never leaves row 4, so its queen is locked to that row.
Amber covers 4 squares, spanning 2 rows and 3 columns (rows 1–2, columns 3–5).
Indigo occupies 5 squares within rows 1–2, columns 7–10.
Sea green: 5 squares across rows 8–10, columns 1–3.
Khaki covers 8 squares, spanning 5 rows and 2 columns (rows 5–9, columns 1–2).
Slate occupies 9 squares within rows 1–5, columns 5–7.
Blue: 12 squares across rows 1–4, columns 1–4.
Orange covers 12 squares, spanning 4 rows and 5 columns (rows 7–10, columns 5–9).
Red occupies 16 squares within rows 4–10, columns 3–6.
Pale yellow: 28 squares across rows 2–10, columns 6–10.
Where to start on this board
Begin where the choices are fewest. On this board that means mint, which owns just 1 square inside a 1×1 window running from row 4 to row 4 and column 10 to column 10. 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.
Work outward to the other flat regions next. Amber and indigo 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 #33
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 6 are the narrowest, touched by only 3 colors. Columns 1, 2, 4, 8, 9 and 10 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 9 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.
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 #33 is rated hard
The rating reflects how much the board concedes up front. Puzzle #33 has two regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 27 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
Stuck on #33 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 #33 is row 3.
A non-spoiling distribution check: solved, #33 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 #33?
Puzzle #33 is rated hard. Its smallest color region holds 1 square, 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 #33 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #33, 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 #33?
Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.