10x10 Queens Puzzle #100
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 18-step reasoning is written out below.
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Step-by-step solution for puzzle #100
What follows is a complete solve of puzzle #100, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 10 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: last square in a region, region confined to one line, line confined to one region, pairing argument on rows, last square in a column, region shadow, pairing argument on columns.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Only row 9, column 2 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 2, the rest of the orange region and every square adjacent to it — diagonals included.
Step 2. The pale yellow region is down to one open square, at row 7, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 1, the remaining pale yellow squares, and the ring of neighbours around row 7, column 1.
Step 3. Blue cannot leave row 3 — every one of its open squares (3 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 7 squares in all.
Step 4. Every open square left in column 3 (5 open squares) belongs to the amber region. Since that column must contain a queen, amber is the colour that supplies it — so amber squares anywhere else on the board are out, 8 squares in all.
Step 5. Khaki and sea green are now boxed into rows 8 and 10 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 5 squares at once.
The board after step 5 — 2 of 10 queens placed. Step 6. Column 4 now has exactly one open square, row 10, column 4, in the sea green region. Columns must each hold a queen, so that placement is forced. That clears row 10, column 4, the rest of the sea green region and every square adjacent to it — diagonals included.
Step 7. Only row 8, column 9 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 9, in khaki and in the ring around that square is now dead.
Step 8. 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: 2 squares come off.
Step 9. The red region has 4 squares left, spread across rows 5 and 6, columns 5–7. 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.
The board after step 9 — 4 of 10 queens placed. Step 10. Here is the pairing argument. The indigo and slate regions have no open squares outside columns 8 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: 2 squares come off in one stroke.
Step 11. Only row 3, column 7 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 7, in blue and in the ring around that square is now dead.
Step 12. The mint region has 2 squares left, spread across rows 1 and 4, 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 13. Red cannot leave row 6 — every one of its open squares (2 open squares) sits there. That row belongs to red, so every square on it owned by another colour is dead: 3 squares in all.
Step 14. 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: column 8 is down to one square, putting a queen on row 1, column 8; mint has one square left, so its queen takes row 4, column 5; slate has one square left, so its queen takes row 5, column 10; amber has one square left, so its queen takes row 2, column 3; red has one square left, so its queen takes row 6, 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 6, column 6.
The board after step 14 — 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 #100
This is puzzle number 100 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.
As with every puzzle in this collection, #100 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.
How the colors divide this board
From smallest to largest, the regions on puzzle #100 are: orange (1 square), blue (3 squares), indigo (7 squares), pale yellow (9 squares), khaki (9 squares), sea green (10 squares), mint (11 squares), red (11 squares), slate (14 squares), amber (25 squares). An even split would hand every color 10 squares, so the orange region is running 9 squares under that average while the amber region is the roomiest area on the grid with 25 squares.
Every color takes one queen and one only, whatever its size — which is why the 24 square difference between orange and amber 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. Blue is the densest region on this grid at 100% of a 1×3 frame; red is the loosest, occupying only 44% of its 5×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Orange covers 1 square, spanning 1 row and 1 column (rows 9–9, columns 2–2). It never leaves row 9, so its queen is locked to that row.
Blue occupies 3 squares within rows 3–3, columns 7–9. It never leaves row 3, so its queen is locked to that row.
Indigo: 7 squares across rows 1–3, columns 8–10.
Pale yellow covers 9 squares, spanning 6 rows and 2 columns (rows 5–10, columns 1–2).
Khaki occupies 9 squares within rows 7–10, columns 6–10.
Sea green: 10 squares across rows 8–10, columns 3–8.
Mint covers 11 squares, spanning 5 rows and 4 columns (rows 1–5, columns 5–8).
Red occupies 11 squares within rows 5–9, columns 3–7.
Slate: 14 squares across rows 4–8, columns 7–10.
Amber covers 25 squares, spanning 6 rows and 5 columns (rows 1–6, columns 1–5).
Where to start on this board
Orange is the natural opening on #100: it is the smallest region on the grid at 1 square, boxed into rows 9–9 and columns 2–2. 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 orange never leaves row 9, its queen must be in that row — and that instantly settles row 9 for everyone else. Sweep along it and X out every square that is not orange. 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. Blue 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 #100
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 4 and 10 are the narrowest, touched by only 3 colors. Column 1 is the tightest column, 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 9, meeting 5 colors, is the opposite case — the busiest line on #100 and the last one you should expect to crack.
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.
Two checks, run alternately, will carry you through the 100 squares of this grid. First: has any of the ten regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 10x10 boards almost always run one of those checks and forget the other.
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 #100 is rated medium
Ratings track how generous the opening is. On #100 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 24-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
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.
A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.
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 #100 is column 1.
Here is a shape check that gives nothing away: when #100 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 #100?
Puzzle #100 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 #100 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #100, 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 #100?
Start with the orange region — the most constrained color on this board, with 1 square spanning rows 9–9 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.