10x10 Queens Puzzle #142

About 10x10 Queens puzzle #142

Board #142 of the 10x10 bank carries a easy 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.

Like every board on this site, #142 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 #142 are: red (2 squares), pale yellow (3 squares), khaki (4 squares), indigo (6 squares), blue (7 squares), slate (7 squares), sea green (9 squares), amber (12 squares), mint (23 squares), orange (27 squares). An even split would hand every color 10 squares, so the red region is running 8 squares under that average while the orange region is the roomiest area on the grid with 27 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 red, with 2 squares, is under far more pressure than orange with 27. A 25-square gap like this one rewards players who start where the choices are fewest.

Square count is only half the story — how a color is arranged counts too. Red is the tidiest shape on #142, packing 100% of its 1×2 bounding box, whereas blue straggles across its 5×3 box at just 47%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

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 easy board.

Red covers 2 squares, spanning 1 row and 2 columns (rows 6–6, columns 4–5). It never leaves row 6, so its queen is locked to that row.

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

Khaki: 4 squares across rows 6–8, columns 8–9.

Indigo covers 6 squares, spanning 4 rows and 2 columns (rows 1–4, columns 1–2).

Blue occupies 7 squares within rows 1–5, columns 1–3.

Slate: 7 squares across rows 5–7, columns 1–4.

Sea green covers 9 squares, spanning 2 rows and 6 columns (rows 9–10, columns 5–10).

Amber occupies 12 squares within rows 1–4, columns 3–7.

Mint: 23 squares across rows 1–5, columns 5–10.

Orange covers 27 squares, spanning 5 rows and 8 columns (rows 6–10, columns 1–8).

Where to start on this board

The most constrained color is where a solve should begin, and on #142 that is red with 2 squares. It reaches from row 6 to row 6 and from column 4 to column 5 — 1 row by 2 columns. 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 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. Sea green 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 #142

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 9 and 10 are the narrowest, touched by only 2 colors. Columns 9 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 6, meeting 5 colors, is the opposite case — the busiest line on #142 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.

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 #142 is rated easy

The easy label comes out of four measurable things about #142: two of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 2 colors, and its largest and smallest regions differ by 25. 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.

If #142 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.

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 #142, start hunting for it near row 9.

Here is a shape check that gives nothing away: when #142 is solved, 3 queens sit on the outer edge of the grid and 7 sit inside it, with 1 queen in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.