10x10 Queens Puzzle #93
Difficulty: medium. 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 20-step reasoning is written out below.
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Step-by-step solution for puzzle #93
What follows is a complete solve of puzzle #93, worked the way a person works it rather than the way a computer does. It runs to 20 deductions — 10 queen placements and 10 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 columns.
Techniques this board needs: region confined to one line, region shadow, last square in a region, pairing argument on rows, pairing argument on columns, last square in a row.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Blue cannot leave row 3 — every one of its open squares (2 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 12 squares in all.
Step 2. Amber is down to 4 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. 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: 8 squares in all.
Step 4. Wherever orange ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
The board after step 4 — 0 of 10 queens placed. Step 5. Only row 10, column 6 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 6, the rest of the sea green region and every square adjacent to it — diagonals included.
Step 6. The orange region is down to one open square, at row 9, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 4, the remaining orange squares, and the ring of neighbours around row 9, column 4.
Step 7. 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.
Step 8. Pale yellow is down to 4 candidates in rows 6–8, columns 1, 2, 3, and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 9. 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 4 squares at once.
The board after step 9 — 2 of 10 queens placed. Step 10. Here is the pairing argument. The amber and mint regions have no open squares outside columns 9 and 10 — 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: 6 squares come off in one stroke.
Step 11. Red cannot leave column 8 — 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: 5 squares in all.
Step 12. All of the khaki region's remaining squares (2 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 13. 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: row 2 is down to one square, putting a queen on row 2, column 10; mint has one square left, so its queen takes row 4, column 9; red has one square left, so its queen takes row 6, column 8; khaki has one square left, so its queen takes row 8, column 7; pale yellow has one square left, so its queen takes row 7, column 5; indigo has one square left, so its queen takes row 1, column 3; blue has one square left, so its queen takes row 3, column 2; slate has one square left, so its queen takes row 5, 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 5, column 1.
The board after step 13 — 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 #93
Puzzle #93 is one of the medium-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, #93 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 #93 are: blue (2 squares), mint (2 squares), sea green (2 squares), orange (4 squares), amber (9 squares), indigo (12 squares), red (14 squares), pale yellow (15 squares), khaki (17 squares), slate (23 squares). An even split would hand every color 10 squares, so the blue region is running 8 squares under that average while the slate region is the roomiest area on the grid with 23 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 blue, with 2 squares, is under far more pressure than slate with 23. A 21-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 blue is the most solid, filling 100% of its 1×2 box, and pale yellow the most diffuse at 50% of its 5×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #93 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Blue: 2 squares across rows 3–3, columns 2–3. It never leaves row 3, so its queen is locked to that row.
Mint covers 2 squares, spanning 1 row and 2 columns (rows 4–4, columns 9–10). It never leaves row 4, so its queen is locked to that row.
Sea green occupies 2 squares within rows 10–10, columns 5–6. It never leaves row 10, so its queen is locked to that row.
Orange: 4 squares across rows 9–10, columns 2–4.
Amber covers 9 squares, spanning 3 rows and 5 columns (rows 1–3, columns 6–10).
Indigo occupies 12 squares within rows 1–3, columns 3–9.
Red: 14 squares across rows 4–8, columns 6–10.
Pale yellow covers 15 squares, spanning 5 rows and 6 columns (rows 6–10, columns 1–6).
Khaki occupies 17 squares within rows 5–10, columns 6–10.
Slate: 23 squares across rows 1–7, columns 1–5.
Where to start on this board
Begin where the choices are fewest. On this board that means blue, which owns just 2 squares inside a 1×2 window running from row 3 to row 3 and column 2 to column 3. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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. Mint, sea green and orange 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 #93
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 4, 5, 8 and 9 are the narrowest, touched by only 3 colors. Column 1 is the tightest column, with 2 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 10 sits at the far end with 4 colors crossing it; it will almost certainly be among the last to settle.
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 #93 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers three single-line regions, a 2-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 21. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a medium board from the rest.
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 #93, start hunting for it near column 1.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 6 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #93?
Puzzle #93 is rated medium. 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 #93 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #93, 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 #93?
Start with the blue region — the most constrained color on this board, with 2 squares spanning rows 3–3 and columns 2–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.