10x10 Queens Puzzle #94
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 24-step reasoning is written out below.
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Step-by-step solution for puzzle #94
What follows is a complete solve of puzzle #94, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 10 queen placements and 14 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 shadow, region confined to one line, pairing argument on rows, pairing argument on columns, last square in a region, line confined to one region, last square in a column.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Wherever slate 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 2. Khaki is down to 3 candidates in rows 8 and 9, columns 1 and 2, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about khaki itself.
Step 3. Sea green cannot leave row 10 — every one of its open squares (2 open squares) sits there. That row belongs to sea green, so every square on it owned by another colour is dead: 9 squares in all.
Step 4. All of the khaki region's remaining squares (2 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 7 squares come off.
Step 5. Mint is down to 4 candidates in rows 3 and 4, columns 2–4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.
The board after step 5 — 0 of 10 queens placed. Step 6. The orange region has 3 squares left, spread across rows 8 and 9, columns 7 and 8. 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 7. Khaki and orange are now boxed into rows 8 and 9 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 10 squares at once.
Step 8. Red is down to 4 candidates in rows 5–7, columns 2 and 3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 9. Red and sea green are now boxed into columns 2 and 3 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 8 squares at once.
Step 10. The mint region is down to one open square, at row 4, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 4, the remaining mint squares, and the ring of neighbours around row 4, column 4.
Step 11. Slate cannot leave row 5 — every one of its open squares (2 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 4 squares in all.
The board after step 11 — 1 of 10 queens placed. Step 12. Here is the pairing argument. The red and pale yellow regions have no open squares outside rows 6 and 7 — 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: 6 squares come off in one stroke.
Step 13. Wherever blue 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 14. Every open square left in row 3 (4 open squares) belongs to the blue region. Since that row must contain a queen, blue is the colour that supplies it — so blue squares anywhere else on the board are out, 1 square in all.
Step 15. Column 5 has only indigo squares left (2 open squares). The column needs a queen and only indigo can provide one, which pins that colour to this column and kills its squares elsewhere: 4 squares.
Step 16. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: column 6 is down to one square, putting a queen on row 3, column 6; indigo has one square left, so its queen takes row 1, column 5; amber has one square left, so its queen takes row 2, column 10; slate has one square left, so its queen takes row 5, column 8; orange has one square left, so its queen takes row 9, column 7; pale yellow has one square left, so its queen takes row 7, column 9; red has one square left, so its queen takes row 6, column 2; khaki has one square left, so its queen takes row 8, column 1; sea green has one square left, so its queen takes row 10, column 3. 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 10, column 3.
The board after step 16 — 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 #94
Board #94 of the 10x10 bank carries a medium rating. Its 100 squares are shared between ten colored regions, and the solution places ten queens: one per row, one per column, one per region, none of them side by side or corner to corner.
Puzzle #94 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #94 are: khaki (3 squares), sea green (3 squares), slate (5 squares), amber (6 squares), mint (6 squares), orange (10 squares), pale yellow (12 squares), red (15 squares), indigo (16 squares), blue (24 squares). An even split would hand every color 10 squares, so the khaki region is running 7 squares under that average while the blue region is the roomiest area on the grid with 24 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 khaki, with 3 squares, is under far more pressure than blue with 24. A 21-square gap like this one rewards players who start where the choices are fewest.
Shape matters as much as size. The sea green region is the most compact color here, filling 100% of the 1×3 box that encloses it, while blue is the most scattered at 43% of its 8×7 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.
Region by region
Here is the full footprint of every color on puzzle #94, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Khaki covers 3 squares, spanning 2 rows and 2 columns (rows 8–9, columns 1–2).
Sea green occupies 3 squares within rows 10–10, columns 1–3. It never leaves row 10, so its queen is locked to that row.
Slate: 5 squares across rows 4–5, columns 8–10.
Amber covers 6 squares, spanning 3 rows and 4 columns (rows 1–3, columns 7–10).
Mint occupies 6 squares within rows 3–4, columns 1–4.
Orange: 10 squares across rows 8–10, columns 4–10.
Pale yellow covers 12 squares, spanning 5 rows and 4 columns (rows 5–9, columns 7–10).
Red occupies 15 squares within rows 5–9, columns 1–5.
Indigo: 16 squares across rows 1–3, columns 1–8.
Blue covers 24 squares, spanning 8 rows and 7 columns (rows 2–9, columns 3–9).
Where to start on this board
The most constrained color is where a solve should begin, and on #94 that is khaki with 3 squares. It reaches from row 8 to row 9 and from column 1 to column 2 — 2 rows by 2 columns. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
With 2 rows and 2 columns of room, khaki needs narrowing rather than solving outright. Test its 3 squares against the neighbours: a candidate that would strand another color is dead, and eliminating even two of them tightens the whole corner of the board around rows 8–9.
Work outward to the other flat regions next. Sea green, mint and slate 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 #94
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 10 are the narrowest, touched by only 2 colors. Column 6 is the tightest column, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 9 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
With ten queens to seat and ten regions competing over 100 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 10x10 boards quickly and players who stall halfway.
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 #94 is rated medium
The rating reflects how much the board concedes up front. Puzzle #94 has one region confined to a single row or column, its smallest color holds 3 squares, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 21 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.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
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 #94, start hunting for it near row 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 #94?
Puzzle #94 is rated medium. Its smallest color region holds 3 squares, its tightest row is touched by 2 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #94 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #94, 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 #94?
Start with the khaki region — the most constrained color on this board, with 3 squares spanning rows 8–9 and columns 1–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.