10x10 Queens Puzzle #190
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 #190
What follows is a complete solve of puzzle #190, 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 proof by contradiction.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, last square in a row, pairing argument on columns, proof by contradiction.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Only row 4, column 2 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 2, the rest of the mint region and every square adjacent to it — diagonals included.
Step 2. All of the blue 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: 10 squares come off.
Step 3. The orange region has 8 squares left, spread across rows 8–10, columns 5–9. 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 sea green 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 — 1 of 10 queens placed. Step 5. Indigo, amber, and blue are now boxed into rows 1–3 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 6 squares at once.
Step 6. The slate region is down to one open square, at row 5, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 7, the remaining slate squares, and the ring of neighbours around row 5, column 7.
Step 7. Look along row 3: one square survives, row 3, column 4, which belongs to blue. A row cannot be left empty, so the queen goes there. That clears row 3, column 4, the rest of the blue region and every square adjacent to it — diagonals included.
Step 8. 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: 5 squares come off in one stroke.
Step 9. The pale yellow region has 4 squares left, spread across rows 6 and 7, columns 8–10. 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 — 3 of 10 queens placed. Step 10. Here is the pairing argument. The indigo and khaki regions have no open squares outside columns 1 and 3 — 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: 4 squares come off in one stroke.
Step 11. Red is down to 3 candidates in rows 6 and 7, columns 5 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 12. Row 8 has a single open square left, at column 1 in the khaki region. Every row needs a queen, so this one is forced. Cross out the whole of row 8 and column 1, the remaining khaki squares, and the ring of neighbours around row 8, column 1.
Step 13. Only row 1, column 3 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 3, the rest of the indigo region and every square adjacent to it — diagonals included.
The board after step 13 — 5 of 10 queens placed. Step 14. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 10. Follow the forced moves that would create and the sea green region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 15. Only row 2, column 6 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 6, in amber and in the ring around that square is now dead.
Step 16. All of the red region's remaining squares (2 open squares) lie in column 5. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 17. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: orange has one square left, so its queen takes row 9, column 8; sea green has one square left, so its queen takes row 10, column 10; pale yellow has one square left, so its queen takes row 7, column 9; red has one square left, so its queen takes row 6, column 5. 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 5.
The board after step 17 — 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 #190
Board #190 of the 10x10 bank carries a medium 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.
As with every puzzle in this collection, #190 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 #190 are: mint (1 square), sea green (4 squares), blue (5 squares), khaki (8 squares), indigo (10 squares), slate (10 squares), pale yellow (10 squares), orange (10 squares), amber (13 squares), red (29 squares). An even split would hand every color 10 squares, so the mint region is running 9 squares under that average while the red region is the roomiest area on the grid with 29 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So mint, with 1 square, is under far more pressure than red with 29. A 28-square gap like this one rewards players who start where the choices are fewest.
Square count is only half the story — how a color is arranged counts too. Mint is the tidiest shape on #190, packing 100% of its 1×1 bounding box, whereas red straggles across its 7×9 box at just 46%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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.
Mint covers 1 square, spanning 1 row and 1 column (rows 4–4, columns 2–2). It never leaves row 4, so its queen is locked to that row.
Sea green occupies 4 squares within rows 9–10, columns 8–10.
Blue: 5 squares across rows 2–4, columns 3–4.
Khaki covers 8 squares, spanning 3 rows and 3 columns (rows 8–10, columns 1–3).
Indigo occupies 10 squares within rows 1–4, columns 1–4.
Slate: 10 squares across rows 2–5, columns 7–10.
Pale yellow covers 10 squares, spanning 4 rows and 3 columns (rows 5–8, columns 8–10).
Orange occupies 10 squares within rows 8–10, columns 4–9.
Amber: 13 squares across rows 1–4, columns 5–10.
Red covers 29 squares, spanning 7 rows and 9 columns (rows 3–9, columns 1–9).
Where to start on this board
The most constrained color is where a solve should begin, and on #190 that is mint with 1 square. It reaches from row 4 to row 4 and from column 2 to column 2 — 1 row 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 mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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. 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 #190
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 6 and 7 are the narrowest, touched by only 2 colors. Columns 1, 5 and 6 are the tightest columns, 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 4 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.
The whole middle game reduces to two sweeps. Sweep the ten colors looking for one with a single square left; sweep the ten rows and ten columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 10x10 solve from a stalled one.
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 #190 is rated medium
Ratings track how generous the opening is. On #190 you get two regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 28-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
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 medium board from the rest.
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.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #190, start hunting for it near row 1.
A non-spoiling distribution check: solved, #190 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 #190?
Puzzle #190 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #190 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #190, 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 #190?
Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.