10x10 Queens Puzzle #155
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 17-step reasoning is written out below.
Previous puzzle · All 10x10 puzzles · Next puzzle
Step-by-step solution for puzzle #155
What follows is a complete solve of puzzle #155, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 10 queen placements and 7 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 region shadow.
Techniques this board needs: last square in a region, region shadow, region confined to one line.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Only row 1, column 1 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 1, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The amber region is down to one open square, at row 2, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 10, the remaining amber squares, and the ring of neighbours around row 2, column 10.
Step 3. The pale yellow region has 3 squares left, spread across rows 7 and 8, columns 8 and 9. 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 4. All of the orange region's remaining squares (2 open squares) lie in row 9. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 9 squares come off.
The board after step 4 — 2 of 10 queens placed. Step 5. Pale yellow cannot leave column 9 — every one of its open squares (2 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 4 squares in all.
Step 6. The sea green region is down to one open square, at row 10, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 6, the remaining sea green squares, and the ring of neighbours around row 10, column 6.
Step 7. Only row 9, column 8 is still open in orange, and since each colour must take a queen, that settles it. Everything else in row 9, in column 8, in orange and in the ring around that square is now dead.
Step 8. The pale yellow region is down to one open square, at row 7, column 9. Every colour holds exactly one queen, so that square is forced. Row 7 and column 9 are now spoken for, along with the squares touching that queen.
Step 9. Slate cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 3 squares in all.
The board after step 9 — 5 of 10 queens placed. Step 10. All of the red region's remaining squares (2 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 11. Khaki cannot leave row 8 — every one of its open squares (3 open squares) sits there. That row belongs to khaki, so every square on it owned by another colour is dead: 1 square in all.
Step 12. The slate region is down to one open square, at row 5, column 2. Every colour holds exactly one queen, so that square is forced. Row 5 and column 2 are now spoken for, along with the squares touching that queen.
Step 13. Only row 6, column 7 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 7, the rest of the red region and every square adjacent to it — diagonals included.
The board after step 13 — 7 of 10 queens placed. Step 14. Blue is down to 2 candidates in rows 3 and 4, columns 3 and 4, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about blue itself.
Step 15. Only row 4, column 5 is still open in mint, and since each colour must take a queen, that settles it. Everything else in row 4, in column 5, in mint and in the ring around that square is now dead.
Step 16. The blue region is down to one open square, at row 3, column 3. Every colour holds exactly one queen, so that square is forced. Row 3 and column 3 are now spoken for, along with the squares touching that queen.
Step 17. Only row 8, column 4 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 4, the rest of the khaki region and every square adjacent to it — diagonals included.
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 #155
10x10 puzzle #155 is rated medium. Ten colored regions cover its 100 squares, and your job is to seat ten queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Like every board on this site, #155 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 #155 are: indigo (1 square), orange (2 squares), amber (4 squares), sea green (5 squares), pale yellow (11 squares), slate (12 squares), khaki (14 squares), red (15 squares), blue (18 squares), mint (18 squares). An even split would hand every color 10 squares, so the indigo region is running 9 squares under that average while the mint 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 mint has 18 to spare, a gap of 17. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; pale yellow is the loosest, occupying only 46% of its 8×3 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Here is the full footprint of every color on puzzle #155, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Indigo occupies 1 square within rows 1–1, columns 1–1. It never leaves row 1, so its queen is locked to that row.
Orange: 2 squares across rows 9–9, columns 7–8. It never leaves row 9, so its queen is locked to that row.
Amber covers 4 squares, spanning 2 rows and 3 columns (rows 1–2, columns 8–10).
Sea green occupies 5 squares within rows 9–10, columns 6–9.
Pale yellow: 11 squares across rows 3–10, columns 8–10.
Slate covers 12 squares, spanning 6 rows and 4 columns (rows 4–9, columns 1–4).
Khaki occupies 14 squares within rows 7–10, columns 1–6.
Red: 15 squares across rows 2–8, columns 7–9.
Blue covers 18 squares, spanning 6 rows and 4 columns (rows 1–6, columns 1–4).
Mint occupies 18 squares within rows 1–7, columns 4–7.
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 1 and 1. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #155.
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. Orange, amber 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 #155
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. Columns 5 and 10 are the tightest columns, 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 9 is the crowded extreme at 5 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.
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 #155 is rated medium
The rating reflects how much the board concedes up front. Puzzle #155 has three regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 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.
Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.
If #155 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.
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 column 5 on this board.
A non-spoiling distribution check: solved, #155 carries 3 queens on the perimeter against 7 queens inside, 1 of them cornered. 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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 10x10 Queens puzzle #155?
Puzzle #155 is rated medium. Its smallest color region holds 1 square, 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 #155 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #155, 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 #155?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.