10x10 Queens Puzzle #299

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 15-step reasoning is written out below.

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Step-by-step solution for puzzle #299

What follows is a complete solve of puzzle #299, worked the way a person works it rather than the way a computer does. It runs to 15 deductions — 10 queen placements and 5 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: region confined to one line, region shadow, pairing argument on columns, last square in a region, line confined to one region, last square in a row.

Show the full walkthrough (7 steps) — contains the solution

  1. Step 1. Red cannot leave column 3 — every one of its open squares (3 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 9 squares in all.

  2. Step 2. Khaki is down to 4 candidates in rows 8–10, columns 1 and 2, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about khaki itself.

  3. Step 3. Mint and khaki are now boxed into columns 1 and 2 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 8 squares at once.

  4. Step 4. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 7 squares come off.

    Puzzle #299 after step 4: 0 of 10 queens placed
    The board after step 4 — 0 of 10 queens placed.

  5. Step 5. Only row 2, column 9 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 9, the rest of the amber region and every square adjacent to it — diagonals included.

  6. Step 6. Every open square left in row 3 (4 open squares) belongs to the blue region. Since that row must contain a queen, blue is the colour that supplies it — so blue squares anywhere else on the board are out, 10 squares in all.

  7. Step 7. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 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; khaki has one square left, so its queen takes row 8, column 2; red has one square left, so its queen takes row 6, column 3; pale yellow has one square left, so its queen takes row 7, column 5; indigo has one square left, so its queen takes row 1, column 4; slate has one square left, so its queen takes row 5, column 10; orange has one square left, so its queen takes row 9, column 6; blue has one square left, so its queen takes row 3, column 7; sea green has one square left, so its queen takes row 10, column 8. 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 8.

    Puzzle #299 after step 7: 10 of 10 queens placed
    The board after step 7 — 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 #299

10x10 puzzle #299 is rated medium. Ten colored regions cover its 100 squares, and your job is to seat ten queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

There is exactly one way to finish #299, 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 #299 are: red (3 squares), amber (4 squares), khaki (4 squares), slate (7 squares), pale yellow (7 squares), indigo (10 squares), mint (10 squares), sea green (12 squares), orange (16 squares), blue (27 squares). An even split would hand every color 10 squares, so the red region is running 7 squares under that average while the blue region is the roomiest area on the grid with 27 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Red has only 3 places to put its queen; blue has 27. The spread between them on this board is 24 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Square count is only half the story — how a color is arranged counts too. Red is the tidiest shape on #299, packing 100% of its 3×1 bounding box, whereas orange straggles across its 5×7 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.

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

Amber: 4 squares across rows 1–2, columns 8–10.

Khaki covers 4 squares, spanning 3 rows and 2 columns (rows 8–10, columns 1–2).

Slate occupies 7 squares within rows 5–7, columns 8–10.

Pale yellow: 7 squares across rows 5–8, columns 4–5.

Indigo covers 10 squares, spanning 4 rows and 5 columns (rows 1–4, columns 1–5).

Mint occupies 10 squares within rows 3–7, columns 1–3.

Sea green: 12 squares across rows 8–10, columns 6–10.

Orange covers 16 squares, spanning 5 rows and 7 columns (rows 6–10, columns 2–8).

Blue occupies 27 squares within rows 1–5, columns 3–10.

Where to start on this board

Open on the tightest color. Here that is red, confined to 3 squares between rows 6 and 8 and columns 3 and 3. Its queen has nowhere else to go, which means those 3 rows and 1 column carry more constraint than anywhere else on #299.

Because red stays inside column 3, its queen belongs to that column and no other region may use it. Cross out the rest of column 3 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. Amber 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 #299

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3, 4, 9 and 10 are the narrowest, touched by only 3 colors. Columns 1, 6 and 7 are the tightest columns, with 3 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 8 meets 5 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

Working through the middle game

After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.

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.

Resist the plausible-looking queen. On #299 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 #299 is rated medium

Ratings track how generous the opening is. On #299 you get one region pinned to a single row or column, a smallest color of 3 squares, a tightest row crossed by 3 colors, and a 24-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.

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.

Stuck on #299 despite the medium 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.

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 #299 the place to look for one is around row 1.

As a sanity check rather than a spoiler: the finished solution to #299 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 #299?

Puzzle #299 is rated medium. 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 #299 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #299, 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 #299?

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