10x10 Queens Puzzle #68
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 20-step reasoning is written out below.
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Step-by-step solution for puzzle #68
What follows is a complete solve of puzzle #68, worked the way a person works it rather than the way a computer does. It runs to 20 deductions — 10 queen placements and 10 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: region shadow, pairing argument on rows, pairing argument on columns, last square in a region, last square in a row, region confined to one line.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Wherever indigo ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 2. Red 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 red itself.
Step 3. The sea green region has 3 squares left, spread across rows 9 and 10, columns 3 and 4. 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. Here is the pairing argument. The khaki, orange, and sea green regions have no open squares outside rows 8–10 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 15 squares come off in one stroke.
The board after step 4 — 0 of 10 queens placed. Step 5. Indigo and orange are now boxed into columns 9 and 10 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 9 squares at once.
Step 6. Here is the pairing argument. The mint, red, khaki, and sea green regions have no open squares outside columns 1–4 — 4 colours confined to 4 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 17 squares come off in one stroke.
Step 7. 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: amber has one square left, so its queen takes row 2, column 5; indigo has one square left, so its queen takes row 1, column 10; orange has one square left, so its queen takes row 9, column 9; row 8 is down to one square, putting a queen on row 8, column 3; sea green has one square left, so its queen takes row 10, 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 10, column 4.
Step 8. The blue region has 4 squares left, spread across rows 3–5, 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.
The board after step 8 — 5 of 10 queens placed. Step 9. Red cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 3 squares in all.
Step 10. All of the pale yellow 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: 4 squares come off.
Step 11. The slate region has 3 squares left, spread across rows 5–7, columns 7 and 8. 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 12. 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 4 is down to one square, putting a queen on row 4, column 2; row 5 is down to one square, putting a queen on row 5, column 7; blue has one square left, so its queen takes row 3, column 8; pale yellow has one square left, so its queen takes row 7, column 6; red has one square left, so its queen takes row 6, 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 6, column 1.
The board after step 12 — 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 #68
This is puzzle number 68 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.
Like every board on this site, #68 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 #68 are: indigo (3 squares), sea green (3 squares), red (4 squares), orange (4 squares), mint (7 squares), khaki (7 squares), amber (12 squares), blue (17 squares), slate (17 squares), pale yellow (26 squares). An even split would hand every color 10 squares, so the indigo region is running 7 squares under that average while the pale yellow region is the roomiest area on the grid with 26 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 3 squares while pale yellow has 26 to spare, a gap of 23. 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 amber is the most solid, filling 80% of its 3×5 box, and pale yellow the most diffuse at 41% of its 8×8 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 #68, 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 3 squares within rows 1–2, columns 9–10.
Sea green: 3 squares across rows 9–10, columns 3–4.
Red covers 4 squares, spanning 3 rows and 2 columns (rows 6–8, columns 1–2).
Orange occupies 4 squares within rows 8–10, columns 9–10.
Mint: 7 squares across rows 3–5, columns 1–4.
Khaki covers 7 squares, spanning 3 rows and 3 columns (rows 8–10, columns 1–3).
Amber occupies 12 squares within rows 1–3, columns 1–5.
Blue: 17 squares across rows 1–5, columns 5–10.
Slate covers 17 squares, spanning 5 rows and 5 columns (rows 4–8, columns 6–10).
Pale yellow occupies 26 squares within rows 3–10, columns 2–9.
Where to start on this board
Indigo is the natural opening on #68: it is the smallest region on the grid at 3 squares, boxed into rows 1–2 and columns 9–10. 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.
Indigo 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.
Work outward to the other flat regions next. Sea green 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 #68
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 6 and 7 are the narrowest, touched by only 3 colors. Columns 5, 6, 7 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 3 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 5 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.
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.
Resist the plausible-looking queen. On #68 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #68 is rated medium
The rating reflects how much the board concedes up front. Puzzle #68 has no region confined to a single row or column, its smallest color holds 3 squares, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 23 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
If #68 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 row 1 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.
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Questions about this puzzle
How hard is 10x10 Queens puzzle #68?
Puzzle #68 is rated medium. Its smallest color region holds 3 squares, its tightest row is touched by 3 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 10x10 puzzle #68 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #68, 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 #68?
Start with the indigo region — the most constrained color on this board, with 3 squares spanning rows 1–2 and columns 9–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.