10x10 Queens Puzzle #98
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 11-step reasoning is written out below.
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Step-by-step solution for puzzle #98
What follows is a complete solve of puzzle #98, worked the way a person works it rather than the way a computer does. It runs to 11 deductions — 10 queen placements and 1 elimination step 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 confined to one line.
Techniques this board needs: last square in a region, region confined to one line, last square in a column.
Show the full walkthrough (3 steps) — contains the solution
Step 1. The board opens by cascading. Each of these placements is forced the moment the one before it is marked, so the whole run is really a single sweep repeated: indigo has one square left, so its queen takes row 1, 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 1; blue has one square left, so its queen takes row 3, column 3; amber has one square left, so its queen takes row 2, column 7. 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 2, column 7.
Step 2. 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: 2 squares in all.
The board after step 2 — 6 of 10 queens placed. Step 3. 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: column 2 is down to one square, putting a queen on row 10, column 2; orange has one square left, so its queen takes row 9, column 6; slate has one square left, so its queen takes row 5, column 5; pale yellow has one square left, so its queen takes row 7, 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 7, column 4.
The board after step 3 — 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 #98
Board #98 of the 10x10 bank carries a hard 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.
As with every puzzle in this collection, #98 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 #98 are: indigo (1 square), khaki (1 square), red (5 squares), mint (6 squares), sea green (9 squares), blue (10 squares), amber (11 squares), orange (18 squares), slate (19 squares), pale yellow (20 squares). An even split would hand every color 10 squares, so the indigo region is running 9 squares under that average while the pale yellow region is the roomiest area on the grid with 20 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 indigo, with 1 square, is under far more pressure than pale yellow with 20. A 19-square gap like this one rewards players who start where the choices are fewest.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #98, packing 100% of its 1×1 bounding box, whereas slate straggles across its 7×7 box at just 39%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a hard board.
Indigo occupies 1 square within rows 1–1, columns 10–10. It never leaves row 1, so its queen is locked to that row.
Khaki: 1 square across rows 8–8, columns 1–1. It never leaves row 8, so its queen is locked to that row.
Red covers 5 squares, spanning 2 rows and 3 columns (rows 5–6, columns 8–10).
Mint occupies 6 squares within rows 1–4, columns 9–10.
Sea green: 9 squares across rows 7–10, columns 1–5.
Blue covers 10 squares, spanning 4 rows and 4 columns (rows 1–4, columns 1–4).
Amber occupies 11 squares within rows 1–4, columns 6–9.
Orange: 18 squares across rows 7–10, columns 6–10.
Slate covers 19 squares, spanning 7 rows and 7 columns (rows 1–7, columns 2–8).
Pale yellow occupies 20 squares within rows 5–9, columns 1–6.
Where to start on this board
The most constrained color is where a solve should begin, and on #98 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 10 to column 10 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Khaki and red 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 #98
A quick way to find the next move is to count how many different colors touch each line. On this board, row 10 is the narrowest, touched by only 2 colors. Columns 5 and 7 are the tightest columns, 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 1 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.
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 #98 is rated hard
The hard label comes out of four measurable things about #98: four of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 19. Boards that score tight on those measures give up their first queen fast.
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.
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 #98, start hunting for it near row 10.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 7 queens in the middle, including 1 in a corner. 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 #98?
Puzzle #98 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #98 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #98, 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 #98?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.