10x10 Queens Puzzle #105

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 #105

What follows is a complete solve of puzzle #105, 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 pairing argument on columns.

Techniques this board needs: last square in a region, region shadow, region confined to one line, line confined to one region, pairing argument on rows, pairing argument on columns.

Show the full walkthrough (11 steps) — contains the solution

  1. Step 1. Only row 6, column 10 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 10, the rest of the red region and every square adjacent to it — diagonals included.

  2. Step 2. The khaki region is down to one open square, at row 8, column 1. Every colour holds exactly one queen, so that square 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.

  3. Step 3. The amber region has 4 squares left, spread across rows 1–3, columns 8 and 9. 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.

  4. Step 4. The mint region is down to one open square, at row 4, column 8. Every colour holds exactly one queen, so that square is forced. Row 4 and column 8 are now spoken for, along with the squares touching that queen.

    Puzzle #105 after step 4: 3 of 10 queens placed
    The board after step 4 — 3 of 10 queens placed.

  5. Step 5. Amber cannot leave column 9 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 2 squares in all.

  6. Step 6. The pale yellow region has 3 squares left, spread across rows 5 and 7, columns 3–5. 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.

  7. Step 7. Sea green cannot leave row 10 — every one of its open squares (2 open squares) sits there. That row belongs to sea green, so every square on it owned by another colour is dead: 5 squares in all.

    Puzzle #105 after step 7: 3 of 10 queens placed
    The board after step 7 — 3 of 10 queens placed.

  8. Step 8. Every open square left in row 9 (4 open squares) belongs to the orange region. Since that row must contain a queen, orange is the colour that supplies it — so orange squares anywhere else on the board are out, 2 squares in all.

  9. Step 9. Slate and pale yellow are now boxed into rows 5 and 7 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 1 square at once.

  10. Step 10. Here is the pairing argument. The indigo, slate, and sea green regions have no open squares outside columns 2–4 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 6 squares come off in one stroke.

  11. 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: pale yellow has one square left, so its queen takes row 7, column 5; slate has one square left, so its queen takes row 5, column 2; sea green has one square left, so its queen takes row 10, column 3; indigo has one square left, so its queen takes row 1, column 4; amber has one square left, so its queen takes row 2, column 9; blue has one square left, so its queen takes row 3, column 6; orange has one square left, so its queen takes row 9, column 7. 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 9, column 7.

    Puzzle #105 after step 11: 10 of 10 queens placed
    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 #105

Puzzle #105 is one of the hard-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.

There is exactly one way to finish #105, 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 #105 are: red (1 square), khaki (1 square), sea green (6 squares), amber (7 squares), mint (7 squares), indigo (9 squares), pale yellow (9 squares), slate (14 squares), blue (22 squares), orange (24 squares). An even split would hand every color 10 squares, so the red region is running 9 squares under that average while the orange region is the roomiest area on the grid with 24 squares.

Every color takes one queen and one only, whatever its size — which is why the 23 square difference between red and orange 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 red is the most solid, filling 100% of its 1×1 box, and pale yellow the most diffuse at 45% of its 4×5 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 #105, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Red: 1 square across rows 6–6, columns 10–10. It never leaves row 6, so its queen is locked to that row.

Khaki covers 1 square, spanning 1 row and 1 column (rows 8–8, columns 1–1). It never leaves row 8, so its queen is locked to that row.

Sea green occupies 6 squares within rows 8–10, columns 1–3.

Amber: 7 squares across rows 1–3, columns 8–10.

Mint covers 7 squares, spanning 3 rows and 3 columns (rows 4–6, columns 8–10).

Indigo occupies 9 squares within rows 1–3, columns 1–4.

Pale yellow: 9 squares across rows 5–8, columns 3–7.

Slate covers 14 squares, spanning 6 rows and 4 columns (rows 4–9, columns 1–4).

Blue occupies 22 squares within rows 1–5, columns 3–8.

Orange: 24 squares across rows 7–10, columns 4–10.

Where to start on this board

Begin where the choices are fewest. On this board that means red, which owns just 1 square inside a 1×1 window running from row 6 to row 6 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 red never leaves row 6, its queen must be in that row — and that instantly settles row 6 for everyone else. Sweep along it and X out every square that is not red. 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. Khaki 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 #105

A quick way to find the next move is to count how many different colors touch each line. On this board, row 10 is the narrowest, touched by only 2 colors. Columns 2, 5, 6, 7 and 9 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 8 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.

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 #105 is rated hard

The rating reflects how much the board concedes up front. Puzzle #105 has four regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 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.

Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.

Stuck on #105 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 #105 is row 10.

A non-spoiling distribution check: solved, #105 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

  1. The board is an N×N grid divided into N colored regions.
  2. Place exactly one queen (♛) in each row, each column, and each colored region.
  3. No two queens may touch each other — not horizontally, vertically, or diagonally.
  4. 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.
  5. 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 #105?

Puzzle #105 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #105 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #105, 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 #105?

Start with the red region — the most constrained color on this board, with 1 square spanning rows 6–6 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.