10x10 Queens Puzzle #90
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 21-step reasoning is written out below.
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Step-by-step solution for puzzle #90
What follows is a complete solve of puzzle #90, worked the way a person works it rather than the way a computer does. It runs to 21 deductions — 10 queen placements and 11 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, last square in a region, region shadow, pairing argument on rows, pairing argument on columns, last square in a row.
Show the full walkthrough (15 steps) — contains the solution
Step 1. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 10 squares in all.
Step 2. All of the mint region's remaining squares (2 open squares) lie in column 8. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 11 squares come off.
Step 3. Only row 1, column 7 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 7, in indigo and in the ring around that square is now dead.
Step 4. Wherever amber 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 5. Khaki cannot leave column 4 — every one of its open squares (2 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 11 squares in all.
The board after step 5 — 1 of 10 queens placed. Step 6. The amber region has 2 squares left, spread across rows 2 and 3, columns 5 and 6. 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 7. Wherever sea green ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 8. The khaki region is down to one open square, at row 8, column 4. Every colour holds exactly one queen, so that square is forced. Row 8 and column 4 are now spoken for, along with the squares touching that queen.
Step 9. Amber and blue are now boxed into rows 2 and 3 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 7 squares at once.
Step 10. The mint region is down to one open square, at row 4, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 8, the remaining mint squares, and the ring of neighbours around row 4, column 8.
The board after step 10 — 3 of 10 queens placed. Step 11. Red is down to 3 candidates in rows 5 and 6, columns 9 and 10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 12. Here is the pairing argument. The blue and red 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: 5 squares come off in one stroke.
Step 13. Slate, red, and pale yellow are now boxed into rows 5–7 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 1 square at once.
Step 14. Here is the pairing argument. The amber and pale yellow regions have no open squares outside columns 5 and 6 — 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: 2 squares come off in one stroke.
Step 15. 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: sea green has one square left, so its queen takes row 10, column 3; orange has one square left, so its queen takes row 9, column 1; row 7 is down to one square, putting a queen on row 7, column 6; amber has one square left, so its queen takes row 2, column 5; blue has one square left, so its queen takes row 3, column 10; red has one square left, so its queen takes row 6, column 9; 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 15 — 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 #90
Board #90 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, #90 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 #90 are: indigo (2 squares), mint (2 squares), khaki (2 squares), sea green (5 squares), red (7 squares), blue (9 squares), orange (10 squares), amber (11 squares), slate (21 squares), pale yellow (31 squares). An even split would hand every color 10 squares, so the indigo region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 31 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 2 places to put its queen; pale yellow has 31. The spread between them on this board is 29 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×2 box that encloses it, while amber is the most scattered at 55% of its 4×5 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
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 medium board.
Indigo: 2 squares across rows 1–1, columns 7–8. It never leaves row 1, so its queen is locked to that row.
Mint covers 2 squares, spanning 2 rows and 1 column (rows 3–4, columns 8–8). It sits wholly inside column 8, so its queen must be in that column.
Khaki occupies 2 squares within rows 8–9, columns 4–4. It sits wholly inside column 4, so its queen must be in that column.
Sea green: 5 squares across rows 9–10, columns 3–6.
Red covers 7 squares, spanning 4 rows and 3 columns (rows 4–7, columns 8–10).
Blue occupies 9 squares within rows 1–4, columns 7–10.
Orange: 10 squares across rows 7–10, columns 1–3.
Amber covers 11 squares, spanning 4 rows and 5 columns (rows 1–4, columns 3–7).
Slate occupies 21 squares within rows 1–7, columns 1–5.
Pale yellow: 31 squares across rows 4–10, columns 4–10.
Where to start on this board
The most constrained color is where a solve should begin, and on #90 that is indigo with 2 squares. It reaches from row 1 to row 1 and from column 7 to column 8 — 1 row 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.
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. Mint, 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 #90
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 5, 6, 8 and 10 are the narrowest, touched by only 3 colors. Columns 1 and 2 are the tightest columns, with 2 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 4, meeting 6 colors, is the opposite case — the busiest line on #90 and the last one you should expect to crack.
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.
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.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #90 is rated medium
The medium label comes out of four measurable things about #90: three of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 29. Boards that score tight on those measures give up their first queen fast.
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.
If #90 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 #90 the place to look for one is around column 1.
Here is a shape check that gives nothing away: when #90 is solved, 4 queens sit on the outer edge of the grid and 6 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 #90?
Puzzle #90 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 #90 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #90, 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 #90?
Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–1 and columns 7–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.