10x10 Queens Puzzle #198
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 #198
What follows is a complete solve of puzzle #198, 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 rows.
Techniques this board needs: region shadow, region confined to one line, last square in a region, pairing argument on rows.
Show the full walkthrough (14 steps) — contains the solution
Step 1. 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 2. All of the slate region's remaining squares (2 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 12 squares come off.
Step 3. Pale yellow cannot leave column 1 — every one of its open squares (3 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 5 squares in all.
Step 4. The amber region is down to one open square, at row 2, column 3. Every colour holds exactly one queen, so that square is forced. Row 2 and column 3 are now spoken for, along with the squares touching that queen.
Step 5. Blue is down to 3 candidates in rows 1 and 3, columns 6 and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
The board after step 5 — 1 of 10 queens placed. Step 6. The red region has 4 squares left, spread across rows 5–7, columns 4 and 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 7. Wherever khaki 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 8. Here is the pairing argument. The pale yellow, khaki, orange, and sea green regions have no open squares outside rows 7–10 — 4 colours confined to 4 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 8 squares come off in one stroke.
Step 9. Red cannot leave column 4 — 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: 4 squares in all.
The board after step 9 — 1 of 10 queens placed. Step 10. All of the khaki region's remaining squares (2 open squares) lie in column 5. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.
Step 11. Only row 1, column 6 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 6, in indigo and in the ring around that square is now dead.
Step 12. The blue region is down to one open square, at row 3, column 7. Every colour holds exactly one queen, so that square is forced. Row 3 and column 7 are now spoken for, along with the squares touching that queen.
Step 13. Orange cannot leave row 9 — every one of its open squares (2 open squares) sits there. That row belongs to orange, so every square on it owned by another colour is dead: 5 squares in all.
Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: khaki has one square left, so its queen takes row 8, column 5; pale yellow has one square left, so its queen takes row 7, column 1; slate has one square left, so its queen takes row 5, column 2; red has one square left, so its queen takes row 6, column 4; sea green has one square left, so its queen takes row 10, column 10; mint has one square left, so its queen takes row 4, column 9; orange has one square left, so its queen takes row 9, column 8. 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 9, column 8.
The board after step 14 — 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 #198
Board #198 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.
Like every board on this site, #198 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 #198 are: slate (2 squares), blue (5 squares), sea green (6 squares), pale yellow (7 squares), amber (9 squares), orange (9 squares), khaki (10 squares), indigo (11 squares), red (11 squares), mint (30 squares). An even split would hand every color 10 squares, so the slate region is running 8 squares under that average while the mint region is the roomiest area on the grid with 30 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Slate has only 2 places to put its queen; mint has 30. The spread between them on this board is 28 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Square count is only half the story — how a color is arranged counts too. Slate is the tidiest shape on #198, packing 100% of its 2×1 bounding box, whereas sea green straggles across its 4×3 box at just 50%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Below is every color on #198 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.
Slate: 2 squares across rows 5–6, columns 2–2. It sits wholly inside column 2, so its queen must be in that column.
Blue covers 5 squares, spanning 3 rows and 3 columns (rows 1–3, columns 6–8).
Sea green occupies 6 squares within rows 7–10, columns 8–10.
Pale yellow: 7 squares across rows 5–9, columns 1–2.
Amber covers 9 squares, spanning 4 rows and 3 columns (rows 1–4, columns 1–3).
Orange occupies 9 squares within rows 8–10, columns 5–9.
Khaki: 10 squares across rows 8–10, columns 1–5.
Indigo covers 11 squares, spanning 5 rows and 4 columns (rows 1–5, columns 3–6).
Red occupies 11 squares within rows 3–8, columns 3–5.
Mint: 30 squares across rows 1–8, columns 5–10.
Where to start on this board
The most constrained color is where a solve should begin, and on #198 that is slate with 2 squares. It reaches from row 5 to row 6 and from column 2 to column 2 — 2 rows 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 slate stays inside column 2, its queen belongs to that column and no other region may use it. Cross out the rest of column 2 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: pale yellow spans only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #198
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 3 colors. Column 10 is the tightest column, with 2 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 9 rows. By contrast row 8 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
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 #198 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers one single-line region, a 2-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 28. 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.
If #198 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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #198 the place to look for one is around column 10.
As a sanity check rather than a spoiler: the finished solution to #198 puts 3 queens on the outer border and 7 queens in the interior, 1 of them in an actual corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 #198?
Puzzle #198 is rated medium. Its smallest color region holds 2 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 #198 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #198, 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 #198?
Start with the slate region — the most constrained color on this board, with 2 squares spanning rows 5–6 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.