10x10 Queens Puzzle #97
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 22-step reasoning is written out below.
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Step-by-step solution for puzzle #97
What follows is a complete solve of puzzle #97, worked the way a person works it rather than the way a computer does. It runs to 22 deductions — 10 queen placements and 12 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 region shadow.
Techniques this board needs: region shadow, region confined to one line, last square in a region.
Show the full walkthrough (19 steps) — contains the solution
Step 1. Wherever amber ends up among its 3 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 2 and 3, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about khaki itself.
Step 3. Orange cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to orange, so every square in 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 row 8. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.
Step 5. 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: 8 squares in all.
The board after step 5 — 0 of 10 queens placed. Step 6. The orange region is down to one open square, at row 9, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 1, the remaining orange squares, and the ring of neighbours around row 9, column 1.
Step 7. Only row 8, column 3 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 3, in khaki and in the ring around that square is now dead.
Step 8. The sea green region is down to one open square, at row 10, column 4. Every colour holds exactly one queen, so that square is forced. Row 10 and column 4 are now spoken for, along with the squares touching that queen.
Step 9. Slate cannot leave column 2 — every one of its open squares (4 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 2 squares in all.
Step 10. The indigo region is down to one open square, at row 1, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 5, the remaining indigo squares, and the ring of neighbours around row 1, column 5.
The board after step 10 — 4 of 10 queens placed. Step 11. Only row 2, column 10 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 10, in amber and in the ring around that square is now dead.
Step 12. All of the blue region's remaining squares (2 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 13. Wherever mint ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 14. All of the red region's remaining squares (3 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
The board after step 14 — 5 of 10 queens placed. Step 15. The mint region has 3 squares left, spread across rows 4–6, columns 7 and 8. 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. The blue region is down to one open square, at row 3, column 6. Every colour holds exactly one queen, so that square is forced. Row 3 and column 6 are now spoken for, along with the squares touching that queen.
Step 17. Pale yellow cannot leave row 7 — every one of its open squares (2 open squares) sits there. That row belongs to pale yellow, so every square on it owned by another colour is dead: 2 squares in all.
Step 18. The mint region has 2 squares left, spread across rows 4 and 5, columns 7 and 8. 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 19. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: red has one square left, so its queen takes row 6, column 9; pale yellow has one square left, so its queen takes row 7, column 7; mint has one square left, so its queen takes row 4, column 8; slate has one square left, so its queen takes row 5, column 2. 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 2.
The board after step 19 — 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 #97
Puzzle #97 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.
Like every board on this site, #97 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.
How the colors divide this board
From smallest to largest, the regions on puzzle #97 are: amber (3 squares), khaki (3 squares), orange (3 squares), sea green (4 squares), red (8 squares), indigo (9 squares), mint (10 squares), slate (14 squares), blue (15 squares), pale yellow (31 squares). An even split would hand every color 10 squares, so the amber region is running 7 squares under that average while the pale yellow region is the roomiest area on the grid with 31 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 amber, with 3 squares, is under far more pressure than pale yellow with 31. A 28-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 orange is the most solid, filling 100% of its 3×1 box, and mint the most diffuse at 50% of its 5×4 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
Amber covers 3 squares, spanning 2 rows and 2 columns (rows 1–2, columns 9–10).
Khaki occupies 3 squares within rows 8–9, columns 2–3.
Orange: 3 squares across rows 8–10, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Sea green covers 4 squares, spanning 2 rows and 3 columns (rows 9–10, columns 2–4).
Red occupies 8 squares within rows 4–7, columns 8–10.
Indigo: 9 squares across rows 1–2, columns 1–5.
Mint covers 10 squares, spanning 5 rows and 4 columns (rows 2–6, columns 7–10).
Slate occupies 14 squares within rows 3–7, columns 1–3.
Blue: 15 squares across rows 1–5, columns 3–8.
Pale yellow covers 31 squares, spanning 7 rows and 7 columns (rows 4–10, columns 4–10).
Where to start on this board
Begin where the choices are fewest. On this board that means amber, which owns just 3 squares inside a 2×2 window running from row 1 to row 2 and column 9 to column 10. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Amber 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. Indigo, khaki and sea green 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 #97
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 3, 7, 8 and 10 are the narrowest, touched by only 3 colors. Column 6 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 5 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.
Keep alternating between two views of the board. In the region view you ask which of the ten colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 10x10 board unwinds.
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 #97 is rated hard
Ratings track how generous the opening is. On #97 you get one region pinned to a single row or column, a smallest color of 3 squares, a tightest row crossed by 3 colors, and a 28-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
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.
If #97 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.
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 #97, start hunting for it near column 6.
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 #97?
Puzzle #97 is rated hard. Its smallest color region holds 3 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 #97 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #97, 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 #97?
Start with the amber region — the most constrained color on this board, with 3 squares spanning rows 1–2 and columns 9–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.