10x10 Queens Puzzle #110

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 #110

What follows is a complete solve of puzzle #110, 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 region shadow.

Techniques this board needs: region confined to one line, last square in a region, region shadow, last square in a row.

Show the full walkthrough (10 steps) — contains the solution

  1. Step 1. Khaki cannot leave column 10 — every one of its open squares (3 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 8 squares in all.

  2. Step 2. All of the blue region's remaining squares (3 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 7 squares come off.

  3. Step 3. Only row 10, column 8 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 8, in sea green and in the ring around that square is now dead.

    Puzzle #110 after step 3: 1 of 10 queens placed
    The board after step 3 — 1 of 10 queens placed.

  4. Step 4. All of the pale yellow region's remaining squares (2 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 8 squares come off.

  5. Step 5. Red cannot leave row 6 — every one of its open squares (2 open squares) sits there. That row belongs to red, so every square on it owned by another colour is dead: 8 squares in all.

  6. Step 6. The orange region has 2 squares left, spread across rows 8 and 9, columns 5 and 6. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.

  7. Step 7. 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: pale yellow has one square left, so its queen takes row 7, column 7; red has one square left, so its queen takes row 6, column 5; khaki has one square left, so its queen takes row 8, column 10; orange has one square left, so its queen takes row 9, 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 9, column 6.

    Puzzle #110 after step 7: 5 of 10 queens placed
    The board after step 7 — 5 of 10 queens placed.

  8. Step 8. Amber cannot leave column 4 — 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: 4 squares in all.

  9. Step 9. All of the slate region's remaining squares (2 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 3 squares come off.

  10. Step 10. 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: row 4 is down to one square, putting a queen on row 4, column 1; slate has one square left, so its queen takes row 5, column 3; indigo has one square left, so its queen takes row 1, column 2; amber has one square left, so its queen takes row 2, column 4; blue has one square left, so its queen takes row 3, 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 3, column 9.

    Puzzle #110 after step 10: 10 of 10 queens placed
    The board after step 10 — 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 #110

Board #110 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.

Puzzle #110 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.

How the colors divide this board

From smallest to largest, the regions on puzzle #110 are: khaki (3 squares), mint (6 squares), pale yellow (6 squares), indigo (7 squares), blue (7 squares), sea green (7 squares), amber (9 squares), orange (11 squares), red (14 squares), slate (30 squares). An even split would hand every color 10 squares, so the khaki region is running 7 squares under that average while the slate region is the roomiest area on the grid with 30 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Khaki has only 3 places to put its queen; slate has 30. The spread between them on this board is 27 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 khaki region is the most compact color here, filling 100% of the 3×1 box that encloses it, while red is the most scattered at 47% 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

Below is every color on #110 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

Khaki occupies 3 squares within rows 6–8, columns 10–10. It sits wholly inside column 10, so its queen must be in that column.

Mint: 6 squares across rows 2–5, columns 1–2.

Pale yellow covers 6 squares, spanning 3 rows and 3 columns (rows 6–8, columns 7–9).

Indigo occupies 7 squares within rows 1–4, columns 1–3.

Blue: 7 squares across rows 1–4, columns 9–10.

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

Amber occupies 9 squares within rows 1–3, columns 4–8.

Orange: 11 squares across rows 7–10, columns 4–8.

Red covers 14 squares, spanning 5 rows and 6 columns (rows 2–6, columns 5–10).

Slate occupies 30 squares within rows 3–10, columns 1–6.

Where to start on this board

The most constrained color is where a solve should begin, and on #110 that is khaki with 3 squares. It reaches from row 6 to row 8 and from column 10 to column 10 — 3 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 khaki 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.

No region here is pinned to a shallow band of rows, so turn the same idea sideways: blue and mint span only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.

The tightest lines on puzzle #110

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 5, 9 and 10 are the narrowest, touched by only 3 colors. Column 3 is the tightest column, with 2 colors.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 9 rows. By contrast row 3 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

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.

Two checks, run alternately, will carry you through the 100 squares of this grid. First: has any of the ten regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 10x10 boards almost always run one of those checks and forget the other.

Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.

Why puzzle #110 is rated hard

What earns a board its rating is the shape of its opening. Here that opening offers one single-line region, a 3-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 27. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.

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 #110 the place to look for one is around column 3.

As a sanity check rather than a spoiler: the finished solution to #110 puts 4 queens on the outer border and 6 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.

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 #110?

Puzzle #110 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #110 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #110, 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 #110?

Start with the khaki region — the most constrained color on this board, with 3 squares spanning rows 6–8 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.