10x10 Queens Puzzle #28
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.
Previous puzzle · All 10x10 puzzles · Next puzzle
Step-by-step solution for puzzle #28
What follows is a complete solve of puzzle #28, 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: last square in a region, region confined to one line, region shadow, line confined to one region, last square in a column.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Only row 10, column 10 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 10, the rest of the sea green region and every square adjacent to it — diagonals included.
Step 2. The orange region is down to one open square, at row 9, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 1, the remaining orange squares, and the ring of neighbours around row 9, column 1.
Step 3. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 8 squares in all.
Step 4. 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.
The board after step 4 — 2 of 10 queens placed. Step 5. Blue is down to 4 candidates in rows 2 and 3, columns 2–4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 6. The pale yellow region has 5 squares left, spread across rows 6–8, columns 2, 3, and 5. 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 7. Column 9 has only khaki squares left (7 open squares). The column needs a queen and only khaki can provide one, which pins that colour to this column and kills its squares elsewhere: 8 squares.
Step 8. Column 8 now has exactly one open square, row 2, column 8, in the amber region. Columns must each hold a queen, so that placement is forced. Everything else in row 2, in column 8, in amber and in the ring around that square is now dead.
Step 9. Only row 3, column 2 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 2, the rest of the blue region and every square adjacent to it — diagonals included.
The board after step 9 — 4 of 10 queens placed. Step 10. Column 7 now has exactly one open square, row 6, column 7, in the red region. Columns must each hold a queen, so that placement is forced. That clears row 6, column 7, the rest of the red region and every square adjacent to it — diagonals included.
Step 11. Mint cannot leave row 4 — every one of its open squares (2 open squares) sits there. That row belongs to mint, so every square on it owned by another colour is dead: 2 squares in all.
Step 12. The pale yellow region has 3 squares left, spread across rows 7 and 8, columns 3 and 5. 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. 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 4 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 6; indigo has one square left, so its queen takes row 1, column 5; pale yellow has one square left, so its queen takes row 7, column 3; khaki has one square left, so its queen takes row 8, column 9. 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 8, column 9.
The board after step 13 — 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 #28
This is puzzle number 28 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.
There is exactly one way to finish #28, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these ten regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #28 are: sea green (1 square), indigo (2 squares), orange (3 squares), mint (5 squares), red (9 squares), blue (10 squares), amber (12 squares), pale yellow (14 squares), slate (17 squares), khaki (27 squares). An even split would hand every color 10 squares, so the sea green region is running 9 squares under that average while the khaki region is the roomiest area on the grid with 27 squares.
Every color takes one queen and one only, whatever its size — which is why the 26 square difference between sea green and khaki 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.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×2 box, and red the most diffuse at 50% of its 6×3 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a hard board.
Sea green covers 1 square, spanning 1 row and 1 column (rows 10–10, columns 10–10). It never leaves row 10, so its queen is locked to that row.
Indigo occupies 2 squares within rows 1–1, columns 5–6. It never leaves row 1, so its queen is locked to that row.
Orange: 3 squares across rows 9–10, columns 1–2.
Mint covers 5 squares, spanning 3 rows and 2 columns (rows 2–4, columns 5–6).
Red occupies 9 squares within rows 5–10, columns 5–7.
Blue: 10 squares across rows 1–3, columns 1–4.
Amber covers 12 squares, spanning 4 rows and 5 columns (rows 1–4, columns 6–10).
Pale yellow occupies 14 squares within rows 6–10, columns 1–5.
Slate: 17 squares across rows 3–8, columns 1–4.
Khaki covers 27 squares, spanning 9 rows and 4 columns (rows 2–10, columns 7–10).
Where to start on this board
Sea green is the natural opening on #28: it is the smallest region on the grid at 1 square, boxed into rows 10–10 and columns 10–10. 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 sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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. Indigo and orange 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 #28
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 5 are the narrowest, touched by only 3 colors. Columns 8 and 9 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 10 is the crowded extreme at 5 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.
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 #28 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 #28 is rated hard
The hard label comes out of four measurable things about #28: three of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 26. Boards that score tight on those measures give up their first queen fast.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a hard board from the rest.
Stuck on #28 despite the hard rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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 #28 is column 8.
A non-spoiling distribution check: solved, #28 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 #28?
Puzzle #28 is rated hard. 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 #28 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #28, 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 #28?
Start with the sea green region — the most constrained color on this board, with 1 square spanning rows 10–10 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.