10x10 Queens Puzzle #143

About 10x10 Queens puzzle #143

10x10 puzzle #143 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.

Like every board on this site, #143 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.

How the colors divide this board

From smallest to largest, the regions on puzzle #143 are: indigo (3 squares), slate (4 squares), mint (5 squares), orange (6 squares), pale yellow (7 squares), amber (9 squares), blue (9 squares), sea green (10 squares), khaki (11 squares), red (36 squares). An even split would hand every color 10 squares, so the indigo region is running 7 squares under that average while the red region is the roomiest area on the grid with 36 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 3 places to put its queen; red has 36. The spread between them on this board is 33 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. Blue is the tidiest shape on #143, packing 100% of its 3×3 bounding box, whereas red straggles across its 7×9 box at just 57%. 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.

Indigo occupies 3 squares within rows 1–2, columns 2–3.

Slate: 4 squares across rows 4–5, columns 8–10.

Mint covers 5 squares, spanning 4 rows and 2 columns (rows 1–4, columns 1–2).

Orange occupies 6 squares within rows 7–10, columns 1–2.

Pale yellow: 7 squares across rows 6–10, columns 9–10.

Amber covers 9 squares, spanning 3 rows and 5 columns (rows 1–3, columns 3–7).

Blue occupies 9 squares within rows 1–3, columns 8–10.

Sea green: 10 squares across rows 7–10, columns 6–9.

Khaki covers 11 squares, spanning 3 rows and 5 columns (rows 8–10, columns 2–6).

Red occupies 36 squares within rows 2–8, columns 1–9.

Where to start on this board

Open on the tightest color. Here that is indigo, confined to 3 squares between rows 1 and 2 and columns 2 and 3. Its queen has nowhere else to go, which means those 2 rows and 2 columns carry more constraint than anywhere else on #143.

Indigo spans 2 rows, so it stays open for now — but it is still the shortest list on the board at 3 squares. Work through those squares first and mark the ones that would leave another region homeless. That pruning is what makes the rest of the grid legible.

Work outward to the other flat regions next. Slate 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 #143

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 5 and 6 are the narrowest, touched by only 2 colors. Columns 1, 4, 5, 7 and 10 are the tightest columns, with 3 colors.

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 8, meeting 5 colors, is the opposite case — the busiest line on #143 and the last one you should expect to crack.

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 #143 is rated medium

Ratings track how generous the opening is. On #143 you get no region pinned to a single row or column, a smallest color of 3 squares, a tightest row crossed by 2 colors, and a 33-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.

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.

If #143 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.

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 #143 the place to look for one is around row 5.

Here is a shape check that gives nothing away: when #143 is solved, 4 queens sit on the outer edge of the grid and 6 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.