10x10 Queens Puzzle #16
Difficulty: hard. 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 26-step reasoning is written out below.
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Step-by-step solution for puzzle #16
What follows is a complete solve of puzzle #16, worked the way a person works it rather than the way a computer does. It runs to 26 deductions — 10 queen placements and 16 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: region shadow, pairing argument on columns, region confined to one line, proof by contradiction, last square in a region, pairing argument on rows.
Show the full walkthrough (18 steps) — contains the solution
Step 1. Wherever indigo 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 2. Slate is down to 3 candidates in rows 4 and 5, columns 3 and 4, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about slate itself.
Step 3. The khaki region has 5 squares left, spread across rows 6–8, columns 1–3. 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. Wherever blue ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Blue, khaki, and sea green are now boxed into columns 1–3 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 11 squares at once.
The board after step 5 — 0 of 10 queens placed. Step 6. All of the slate region's remaining squares (2 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 8 squares come off.
Step 7. Test row 1, column 8 for indigo: place a queen there, run the forced consequences through, and the mint region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 8. Here is the pairing argument. The indigo and red regions have no open squares outside columns 9 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: 9 squares come off in one stroke.
Step 9. Test row 1, column 10 for indigo: place a queen there, run the forced consequences through, and the amber region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 9 — 0 of 10 queens placed. Step 10. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 10. Follow the forced moves that would create and the mint region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 11. Test row 3, column 9 for indigo: place a queen there, run the forced consequences through, and the red region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 12. Now a short proof by contradiction. Suppose the indigo queen went to row 3, column 10. Follow the forced moves that would create and the blue region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 13. Indigo cannot leave column 9 — every one of its open squares (2 open squares) sits there. That column belongs to indigo, so every square in it owned by another colour is dead: 1 square in all.
Step 14. Now a short proof by contradiction. Suppose the indigo queen went to row 4, column 9. Follow the forced moves that would create and the amber region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
The board after step 14 — 0 of 10 queens placed. Step 15. Only row 1, column 9 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 9, in indigo and in the ring around that square is now dead.
Step 16. Wherever amber 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 17. Amber, blue, mint, and slate are now boxed into rows 2–5 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 5 squares at once.
Step 18. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: red has one square left, so its queen takes row 6, column 10; pale yellow has one square left, so its queen takes row 7, column 5; khaki has one square left, so its queen takes row 8, column 2; sea green has one square left, so its queen takes row 10, column 1; blue has one square left, so its queen takes row 3, column 3; amber has one square left, so its queen takes row 2, column 7; slate has one square left, so its queen takes row 5, column 4; mint has one square left, so its queen takes row 4, column 6; orange has one square left, so its queen takes row 9, column 8. 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 9, column 8.
The board after step 18 — 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 #16
This is puzzle number 16 in the 10x10 collection, rated hard. 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, #16 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 #16 are: slate (3 squares), red (4 squares), khaki (6 squares), sea green (6 squares), indigo (7 squares), blue (7 squares), mint (8 squares), amber (18 squares), pale yellow (18 squares), orange (23 squares). An even split would hand every color 10 squares, so the slate region is running 7 squares under that average while the orange region is the roomiest area on the grid with 23 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Slate is working with 3 squares while orange has 23 to spare, a gap of 20. 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 orange is the most solid, filling 96% of its 4×6 box, and pale yellow the most diffuse at 37% of its 7×7 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 #16, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Slate covers 3 squares, spanning 2 rows and 2 columns (rows 4–5, columns 3–4).
Red occupies 4 squares within rows 4–6, columns 9–10.
Khaki: 6 squares across rows 5–8, columns 1–3.
Sea green covers 6 squares, spanning 4 rows and 2 columns (rows 7–10, columns 1–2).
Indigo occupies 7 squares within rows 1–4, columns 8–10.
Blue: 7 squares across rows 3–5, columns 1–4.
Mint covers 8 squares, spanning 4 rows and 3 columns (rows 2–5, columns 5–7).
Amber occupies 18 squares within rows 1–3, columns 1–9.
Pale yellow: 18 squares across rows 4–10, columns 3–9.
Orange covers 23 squares, spanning 4 rows and 6 columns (rows 7–10, columns 5–10).
Where to start on this board
Slate is the natural opening on #16: it is the smallest region on the grid at 3 squares, boxed into rows 4–5 and columns 3–4. Since one of those squares must hold a queen, that block of 2 rows and 2 columns is the most contested part of the board.
Slate does not collapse onto a single line, so it will not hand you a queen on move one. What it does give you is a short list of 3 candidates. Take each one in turn, imagine the queen there, cross out its row, its column and the eight squares around it, and ask whether any other region is left with nowhere to go. One or two candidates usually fail that test immediately.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: slate, red and sea green span only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #16
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Column 10 is the tightest column, with 3 colors.
A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 9 rows. Row 5 is the crowded extreme at 6 colors — worth revisiting later, not now.
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.
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 #16 is rated hard
The rating reflects how much the board concedes up front. Puzzle #16 has no region confined to a single row or column, its smallest color holds 3 squares, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 20 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.
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.
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 1 on this board.
A non-spoiling distribution check: solved, #16 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.
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Questions about this puzzle
How hard is 10x10 Queens puzzle #16?
Puzzle #16 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 2 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 10x10 puzzle #16 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #16, 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 #16?
Start with the slate region — the most constrained color on this board, with 3 squares spanning rows 4–5 and columns 3–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.