9x9 Queens Puzzle #90
Difficulty: hard. Place 9 queens on this 9x9 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 17-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 17 deductions — 9 queen placements and 8 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 columns, last square in a region, line confined to one region, last square in a row.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Wherever indigo 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 2. Slate is down to 4 candidates in rows 5 and 6, columns 6–8, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about slate itself.
Step 3. Orange cannot leave column 3 — 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: 11 squares in all.
Step 4. Here is the pairing argument. The blue and pale yellow regions have no open squares outside columns 1 and 2 — 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 5. Only row 4, column 4 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 4, the rest of the mint region and every square adjacent to it — diagonals included.
The board after step 5 — 1 of 9 queens placed. Step 6. 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.
Step 7. Wherever blue 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. Khaki is down to 4 candidates in rows 8 and 9, columns 6–8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 9. Row 2 has only amber squares left (3 open squares). The row needs a queen and only amber can provide one, which pins that colour to this row and kills its squares elsewhere: 4 squares.
The board after step 9 — 2 of 9 queens placed. Step 10. 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: row 3 is down to one square, putting a queen on row 3, column 1; pale yellow has one square left, so its queen takes row 7, column 2; orange has one square left, so its queen takes row 9, column 3; khaki has one square left, so its queen takes row 8, column 6. 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 8, column 6.
Step 11. All of the slate region's remaining squares (2 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 12. Only row 6, column 9 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 9, in red and in the ring around that square is now dead.
Step 13. The slate region is down to one open square, at row 5, column 7. Every colour holds exactly one queen, so that square is forced. Row 5 and column 7 are now spoken for, along with the squares touching that queen.
Step 14. Only row 2, column 8 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 8, the rest of the amber region and every square adjacent to it — diagonals included.
The board after step 14 — 9 of 9 queens placed.
That completes the board: 9 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 9x9 Queens puzzle #90
Board #90 of the 9x9 bank carries a hard rating. Its 81 squares are shared between nine colored regions, and the solution places nine 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: orange (2 squares), indigo (4 squares), slate (4 squares), blue (6 squares), pale yellow (7 squares), mint (11 squares), red (14 squares), khaki (16 squares), amber (17 squares). An even split would hand every color 9 squares, so the orange region is running 7 squares under that average while the amber region is the roomiest area on the grid with 17 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Orange has only 2 places to put its queen; amber has 17. The spread between them on this board is 15 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
A region's outline can constrain it more than its size does. Here orange is the most solid, filling 100% of its 2×1 box, and khaki the most diffuse at 53% of its 5×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
Orange: 2 squares across rows 8–9, columns 3–3. It sits wholly inside column 3, so its queen must be in that column.
Indigo covers 4 squares, spanning 3 rows and 2 columns (rows 1–3, columns 4–5).
Slate occupies 4 squares within rows 5–6, columns 6–8.
Blue: 6 squares across rows 3–6, columns 1–2.
Pale yellow covers 7 squares, spanning 4 rows and 2 columns (rows 6–9, columns 1–2).
Mint occupies 11 squares within rows 1–5, columns 1–4.
Red: 14 squares across rows 4–9, columns 6–9.
Khaki covers 16 squares, spanning 5 rows and 6 columns (rows 5–9, columns 3–8).
Amber occupies 17 squares within rows 1–4, columns 5–9.
Where to start on this board
Open on the tightest color. Here that is orange, confined to 2 squares between rows 8 and 9 and columns 3 and 3. Its queen has nowhere else to go, which means those 2 rows and 1 column carry more constraint than anywhere else on #90.
Because orange stays inside column 3, its queen belongs to that column and no other region may use it. Cross out the rest of column 3 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.
Work outward to the other flat regions next. Slate is no more than two rows tall, so its queen is 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 1, 2 and 7 are the narrowest, touched by only 3 colors. Column 9 is the tightest column, with 2 colors.
A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 8 rows. Row 6 is the crowded extreme at 5 colors — worth revisiting later, not now.
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.
With nine queens to seat and nine regions competing over 81 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 9x9 boards quickly and players who stall halfway.
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 hard
The hard label comes out of four measurable things about #90: one of its regions is 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 15. 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.
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 9x9
At 9x9 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 9.
A non-spoiling distribution check: solved, #90 carries 4 queens on the perimeter against 5 queens inside, with the corners empty. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.
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 9x9 Queens puzzle #90?
Puzzle #90 is rated hard. 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 9x9 puzzle #90 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #90, has been verified to have exactly one valid arrangement of nine 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 orange region — the most constrained color on this board, with 2 squares spanning rows 8–9 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.