10x10 Queens Puzzle #196
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.
Previous puzzle · All 10x10 puzzles · Next puzzle
Step-by-step solution for puzzle #196
What follows is a complete solve of puzzle #196, 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 proof by contradiction.
Techniques this board needs: last square in a region, region confined to one line, region shadow, proof by contradiction, line confined to one region, last square in a row.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Only row 1, column 10 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 10, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The mint region is down to one open square, at row 4, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 9, the remaining mint squares, and the ring of neighbours around row 4, column 9.
Step 3. Only row 2, column 8 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 8, in amber and in the ring around that square is now dead.
Step 4. All of the blue region's remaining squares (2 open squares) lie in row 3. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.
The board after step 4 — 3 of 10 queens placed. Step 5. Khaki cannot leave row 8 — every one of its open squares (2 open squares) sits there. That row belongs to khaki, so every square on it owned by another colour is dead: 9 squares in all.
Step 6. The pale yellow region has 4 squares left, spread across rows 5–7, columns 6 and 7. 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 red ends up among its 7 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 8. Orange is down to 6 candidates in rows 7, 9, and 10, columns 1–5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
Step 9. Test row 3, column 6 for blue: place a queen there, run the forced consequences through, and column 2 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 9 — 3 of 10 queens placed. Step 10. The blue region is down to one open square, at row 3, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 2, the remaining blue squares, and the ring of neighbours around row 3, column 2.
Step 11. Only row 8, column 3 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 3, in khaki and in the ring around that square is now dead.
Step 12. All of the slate region's remaining squares (4 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
Step 13. Orange cannot leave column 5 — 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: 2 squares in all.
The board after step 13 — 5 of 10 queens placed. Step 14. Red is down to 2 candidates in rows 5 and 6, columns 4 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 15. Row 10 has only sea green squares left (3 open squares). The row needs a queen and only sea green can provide one, which pins that colour to this row and kills its squares elsewhere: 2 squares.
Step 16. Now a short proof by contradiction. Suppose the slate queen went to row 6, column 1. Follow the forced moves that would create and the sea green region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 17. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: row 6 is down to one square, putting a queen on row 6, column 4; orange has one square left, so its queen takes row 9, column 5; sea green has one square left, so its queen takes row 10, column 7; pale yellow has one square left, so its queen takes row 7, column 6; slate has one square left, so its queen takes row 5, column 1. 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 1.
The board after step 17 — 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 #196
This is puzzle number 196 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.
Puzzle #196 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #196 are: indigo (1 square), khaki (2 squares), mint (5 squares), amber (8 squares), red (10 squares), orange (10 squares), sea green (11 squares), blue (17 squares), slate (18 squares), pale yellow (18 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 18 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 1 square while pale yellow has 18 to spare, a gap of 17. Attack the cramped end and the roomy regions tend to resolve themselves.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and orange the most diffuse at 42% of its 4×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Here is the full footprint of every color on puzzle #196, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 10–10). It never leaves row 1, so its queen is locked to that row.
Khaki occupies 2 squares within rows 8–8, columns 2–3. It never leaves row 8, so its queen is locked to that row.
Mint: 5 squares across rows 3–6, columns 9–10.
Amber covers 8 squares, spanning 3 rows and 4 columns (rows 1–3, columns 7–10).
Red occupies 10 squares within rows 5–8, columns 3–6.
Orange: 10 squares across rows 7–10, columns 1–6.
Sea green covers 11 squares, spanning 3 rows and 6 columns (rows 8–10, columns 4–9).
Blue occupies 17 squares within rows 1–4, columns 1–7.
Slate: 18 squares across rows 3–9, columns 1–5.
Pale yellow covers 18 squares, spanning 7 rows and 5 columns (rows 4–10, columns 6–10).
Where to start on this board
Indigo is the natural opening on #196: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 10–10. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of 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 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 #196
A quick way to find the next move is to count how many different colors touch each line. On this board, row 2 is the narrowest, touched by only 2 colors. Columns 1 and 8 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 8 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.
With ten queens to seat and ten regions competing over 100 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 10x10 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 #196 is rated medium
The rating reflects how much the board concedes up front. Puzzle #196 has three regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 17 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 2 on this board.
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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 10x10 Queens puzzle #196?
Puzzle #196 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #196 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #196, 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 #196?
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.