10x10 Queens Puzzle #34
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 17-step reasoning is written out below.
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Step-by-step solution for puzzle #34
What follows is a complete solve of puzzle #34, 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 line confined to one region.
Techniques this board needs: region shadow, region confined to one line, last square in a region, line confined to one region, last square in a row.
Show the full walkthrough (11 steps) — contains the solution
Step 1. Wherever indigo 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 2. Amber is down to 3 candidates in rows 1 and 2, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 3. Orange cannot leave column 10 — 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: 10 squares in all.
Step 4. The indigo region is down to one open square, at row 1, column 9. Every colour holds exactly one queen, so that square is forced. Row 1 and column 9 are now spoken for, along with the squares touching that queen.
The board after step 4 — 1 of 10 queens placed. Step 5. Only row 2, column 1 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 1, the rest of the amber region and every square adjacent to it — diagonals included.
Step 6. 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. Cross out the whole of row 3 and column 3, the remaining blue squares, and the ring of neighbours around row 3, column 3.
Step 7. Red cannot leave column 2 — every one of its open squares (3 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 3 squares in all.
The board after step 7 — 3 of 10 queens placed. Step 8. All of the pale yellow region's remaining squares (3 open squares) lie in column 8. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
Step 9. The mint region has 4 squares left, spread across rows 4 and 5, columns 5 and 7. 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 10. Every open square left in row 4 (2 open squares) belongs to the mint region. Since that row must contain a queen, mint is the colour that supplies it — so mint squares anywhere else on the board are out, 2 squares in all.
Step 11. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: row 5 is down to one square, putting a queen on row 5, column 4; mint has one square left, so its queen takes row 4, column 7; row 6 is down to one square, putting a queen on row 6, column 2; row 8 is down to one square, putting a queen on row 8, column 6; sea green has one square left, so its queen takes row 10, column 5; orange has one square left, so its queen takes row 9, column 10; pale yellow has one square left, so its queen takes row 7, 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 7, column 8.
The board after step 11 — 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 #34
Board #34 of the 10x10 bank carries a hard rating. Its 100 squares are shared between ten colored regions, and the solution places ten queens: one per row, one per column, one per region, none of them side by side or corner to corner.
Like every board on this site, #34 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 #34 are: orange (2 squares), amber (3 squares), indigo (4 squares), sea green (8 squares), blue (11 squares), khaki (12 squares), slate (14 squares), red (14 squares), pale yellow (15 squares), mint (17 squares). An even split would hand every color 10 squares, so the orange region is running 8 squares under that average while the mint region is the roomiest area on the grid with 17 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Orange has only 2 places to put its queen; mint has 17. The spread between them on this board is 15 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Shape matters as much as size. The orange region is the most compact color here, filling 100% of the 2×1 box that encloses it, while khaki is the most scattered at 40% of its 5×6 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.
Region by region
Here is the full footprint of every color on puzzle #34, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Orange covers 2 squares, spanning 2 rows and 1 column (rows 9–10, columns 10–10). It sits wholly inside column 10, so its queen must be in that column.
Amber occupies 3 squares within rows 1–2, columns 1–2.
Indigo: 4 squares across rows 1–3, columns 9–10.
Sea green covers 8 squares, spanning 3 rows and 6 columns (rows 8–10, columns 3–8).
Blue occupies 11 squares within rows 1–3, columns 2–7.
Khaki: 12 squares across rows 6–10, columns 2–7.
Slate covers 14 squares, spanning 6 rows and 5 columns (rows 3–8, columns 2–6).
Red occupies 14 squares within rows 3–10, columns 1–3.
Pale yellow: 15 squares across rows 4–10, columns 8–10.
Mint covers 17 squares, spanning 6 rows and 5 columns (rows 1–6, columns 5–9).
Where to start on this board
The most constrained color is where a solve should begin, and on #34 that is orange with 2 squares. It reaches from row 9 to row 10 and from column 10 to column 10 — 2 rows by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
Because orange stays inside column 10, its queen belongs to that column and no other region may use it. Cross out the rest of column 10 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
Work outward to the other flat regions next. Amber 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 #34
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, 5 and 7 are the narrowest, touched by only 4 colors. Column 1 is the tightest column, with 2 colors.
A line with few colors is a line close to resolving. Where only 4 colors appear, one of them owns the row outright and the others must fit into the remaining 9 rows. Row 10 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
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.
Resist the plausible-looking queen. On #34 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 #34 is rated hard
The rating reflects how much the board concedes up front. Puzzle #34 has one region confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 4 colors, and the gap between its largest and smallest regions is 15 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.
If #34 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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #34 the place to look for one is around column 1.
A non-spoiling distribution check: solved, #34 carries 4 queens on the perimeter against 6 queens inside, with the corners empty. 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 #34?
Puzzle #34 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 4 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #34 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #34, 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 #34?
Start with the orange region — the most constrained color on this board, with 2 squares spanning rows 9–10 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.