10x10 Queens Puzzle #5
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 16-step reasoning is written out below.
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Step-by-step solution for puzzle #5
What follows is a complete solve of puzzle #5, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 10 queen placements and 6 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 region shadow.
Techniques this board needs: region confined to one line, region shadow, last square in a region.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Indigo cannot leave column 10 — 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: 10 squares in all.
Step 2. Mint is down to 4 candidates in rows 2–4, columns 1 and 2, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about mint itself.
Step 3. Only row 6, column 1 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 1, in red and in the ring around that square is now dead.
Step 4. The mint region is down to one open square, at row 4, column 2. Every colour holds exactly one queen, so that square is forced. Row 4 and column 2 are now spoken for, along with the squares touching that queen.
The board after step 4 — 2 of 10 queens placed. Step 5. Slate is down to 2 candidates in rows 5 and 7, columns 8 and 9, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about slate itself.
Step 6. All of the khaki region's remaining squares (2 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
Step 7. Only row 5, column 8 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 8, in slate and in the ring around that square is now dead.
Step 8. All of the amber region's remaining squares (2 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.
Step 9. 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: indigo has one square left, so its queen takes row 1, column 10; blue has one square left, so its queen takes row 3, column 5; amber has one square left, so its queen takes row 2, column 7; sea green has one square left, so its queen takes row 10, column 6. 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 6.
The board after step 9 — 7 of 10 queens placed. Step 10. Wherever pale yellow 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 11. The orange region is down to one open square, at row 9, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 3, the remaining orange squares, and the ring of neighbours around row 9, column 3.
Step 12. Only row 7, column 4 is still open in pale yellow, and since each colour must take a queen, that settles it. Everything else in row 7, in column 4, in pale yellow and in the ring around that square is now dead.
Step 13. The khaki region is down to one open square, at row 8, column 9. Every colour holds exactly one queen, so that square is forced. Row 8 and column 9 are now spoken for, along with the squares touching that queen.
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 #5
Puzzle #5 is one of the medium-rated boards in the 10x10 set. Ten colors divide its 100 squares, and you win by placing ten queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
As with every puzzle in this collection, #5 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 #5 are: indigo (2 squares), red (2 squares), mint (4 squares), slate (5 squares), khaki (5 squares), blue (8 squares), orange (13 squares), sea green (16 squares), amber (17 squares), pale yellow (28 squares). An even split would hand every color 10 squares, so the indigo region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 28 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 indigo, with 2 squares, is under far more pressure than pale yellow with 28. A 26-square gap like this one rewards players who start where the choices are fewest.
Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 2×1 box that encloses it, while pale yellow is the most scattered at 52% of its 9×6 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
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 medium board.
Indigo occupies 2 squares within rows 1–2, columns 10–10. It sits wholly inside column 10, so its queen must be in that column.
Red: 2 squares across rows 5–6, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Mint covers 4 squares, spanning 3 rows and 2 columns (rows 2–4, columns 1–2).
Slate occupies 5 squares within rows 5–7, columns 8–10.
Khaki: 5 squares across rows 8–10, columns 9–10.
Blue covers 8 squares, spanning 3 rows and 4 columns (rows 1–3, columns 4–7).
Orange occupies 13 squares within rows 7–10, columns 1–4.
Sea green: 16 squares across rows 5–10, columns 5–9.
Amber covers 17 squares, spanning 6 rows and 5 columns (rows 1–6, columns 6–10).
Pale yellow occupies 28 squares within rows 1–9, columns 1–6.
Where to start on this board
Begin where the choices are fewest. On this board that means indigo, which owns just 2 squares inside a 2×1 window running from row 1 to row 2 and column 10 to column 10. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because indigo 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. Red 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 #5
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 4 and 10 are the narrowest, touched by only 3 colors. Column 3 is the tightest column, 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 6 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
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 #5 is rated medium
The medium label comes out of four measurable things about #5: two of its regions are locked to one row or column, its smallest color spans 2 squares, 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.
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.
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 #5, start hunting for it near column 3.
A non-spoiling distribution check: solved, #5 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 #5?
Puzzle #5 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #5 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #5, 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 #5?
Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.