10x10 Queens Puzzle #36
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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #36
What follows is a complete solve of puzzle #36, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 10 queen placements and 9 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 line confined to one region.
Techniques this board needs: region confined to one line, last square in a region, region shadow, line confined to one region, last square in a row.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Amber cannot leave column 10 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 10 squares in all.
Step 2. The sea green region is down to one open square, at row 10, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 9, the remaining sea green squares, and the ring of neighbours around row 10, column 9.
Step 3. The indigo region has 4 squares left, spread across rows 1–3, 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 4. All of the slate region's remaining squares (2 open squares) lie in column 6. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 11 squares come off.
The board after step 4 — 1 of 10 queens placed. Step 5. Pale yellow is down to 4 candidates in rows 6–8, columns 1 and 2, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 6. The orange region is down to one open square, at row 9, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 2, the remaining orange squares, and the ring of neighbours around row 9, column 2.
Step 7. Wherever red 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. All of the pale yellow region's remaining squares (2 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.
Step 9. Row 8 has only khaki squares left (4 open squares). The row needs a queen and only khaki can provide one, which pins that colour to this row and kills its squares elsewhere: 8 squares.
The board after step 9 — 2 of 10 queens placed. Step 10. Row 7 has a single open square left, at column 1 in the pale yellow region. Every row needs a queen, so this one is forced. Row 7 and column 1 are now spoken for, along with the squares touching that queen.
Step 11. Row 6 has only red squares left (2 open squares). The row needs a queen and only red can provide one, which pins that colour to this row and kills its squares elsewhere: 2 squares.
Step 12. Every open square left in column 3 (4 open squares) belongs to the mint region. Since that column must contain a queen, mint is the colour that supplies it — so mint squares anywhere else on the board are out, 5 squares in all.
Step 13. 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: row 5 is down to one square, putting a queen on row 5, column 6; red has one square left, so its queen takes row 6, column 8; indigo has one square left, so its queen takes row 1, column 7; amber has one square left, so its queen takes row 2, column 10; blue has one square left, so its queen takes row 3, column 5; mint has one square left, so its queen takes row 4, column 3; khaki has one square left, so its queen takes row 8, 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 8, column 4.
The board after step 13 — 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 #36
This is puzzle number 36 in the 10x10 collection, rated medium. The grid holds 100 squares split between ten colored regions, and a finished board carries ten queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
There is exactly one way to finish #36, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these ten regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #36 are: amber (2 squares), slate (2 squares), sea green (3 squares), pale yellow (4 squares), orange (5 squares), indigo (7 squares), blue (8 squares), khaki (20 squares), mint (22 squares), red (27 squares). An even split would hand every color 10 squares, so the amber region is running 8 squares under that average while the red region is the roomiest area on the grid with 27 squares.
Every color takes one queen and one only, whatever its size — which is why the 25 square difference between amber and red is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
Look at arrangement as well as area. Amber is the densest region on this grid at 100% of a 2×1 frame; khaki is the loosest, occupying only 48% of its 6×7 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Below is every color on #36 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.
Amber: 2 squares across rows 1–2, columns 10–10. It sits wholly inside column 10, so its queen must be in that column.
Slate covers 2 squares, spanning 2 rows and 1 column (rows 4–5, columns 6–6). It sits wholly inside column 6, so its queen must be in that column.
Sea green occupies 3 squares within rows 9–10, columns 9–10.
Pale yellow: 4 squares across rows 6–8, columns 1–2.
Orange covers 5 squares, spanning 2 rows and 3 columns (rows 9–10, columns 1–3).
Indigo occupies 7 squares within rows 1–3, columns 7–9.
Blue: 8 squares across rows 1–3, columns 5–7.
Khaki covers 20 squares, spanning 6 rows and 7 columns (rows 5–10, columns 2–8).
Mint occupies 22 squares within rows 1–6, columns 1–5.
Red: 27 squares across rows 3–10, columns 5–10.
Where to start on this board
Amber is the natural opening on #36: it is the smallest region on the grid at 2 squares, boxed into rows 1–2 and columns 10–10. Since one of those squares must hold a queen, that block of 2 rows and 1 column is the most contested part of the board.
Because amber stays inside column 10, its queen belongs to that column and no other region may use it. Cross out the rest of column 10 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, orange 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 #36
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 4, 7 and 8 are the narrowest, touched by only 3 colors. Column 4 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 9 rows. Row 10 is the crowded extreme at 4 colors — worth revisiting later, not now.
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 #36 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 2-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 25. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
Stuck on #36 despite the medium rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #36 is column 4.
A non-spoiling distribution check: solved, #36 carries 4 queens on the perimeter against 6 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 10x10 Queens puzzle #36?
Puzzle #36 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #36 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #36, 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 #36?
Start with the amber region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.