10x10 Queens Puzzle #186

About 10x10 Queens puzzle #186

Board #186 of the 10x10 bank carries a medium 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, #186 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 #186 are: red (2 squares), indigo (3 squares), mint (3 squares), pale yellow (3 squares), sea green (3 squares), slate (9 squares), orange (9 squares), amber (13 squares), blue (20 squares), khaki (35 squares). An even split would hand every color 10 squares, so the red region is running 8 squares under that average while the khaki region is the roomiest area on the grid with 35 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 khaki with 35. A 33-square gap like this one rewards players who start where the choices are fewest.

Shape matters as much as size. The red region is the most compact color here, filling 100% of the 2×1 box that encloses it, while blue is the most scattered at 50% of its 5×8 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

Here is the full footprint of every color on puzzle #186, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Red: 2 squares across rows 5–6, columns 10–10. It sits wholly inside column 10, so its queen must be in that column.

Indigo covers 3 squares, spanning 2 rows and 2 columns (rows 1–2, columns 1–2).

Mint occupies 3 squares within rows 4–5, columns 9–10.

Pale yellow: 3 squares across rows 7–7, columns 4–6. It never leaves row 7, so its queen is locked to that row.

Sea green covers 3 squares, spanning 1 row and 3 columns (rows 10–10, columns 5–7). It never leaves row 10, so its queen is locked to that row.

Slate occupies 9 squares within rows 4–6, columns 1–4.

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

Amber covers 13 squares, spanning 3 rows and 7 columns (rows 1–3, columns 4–10).

Blue occupies 20 squares within rows 1–5, columns 1–8.

Khaki: 35 squares across rows 4–10, columns 2–10.

Where to start on this board

The most constrained color is where a solve should begin, and on #186 that is red with 2 squares. It reaches from row 5 to row 6 and from column 10 to column 10 — 2 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 red 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.

Work outward to the other flat regions next. Pale yellow, sea green, indigo and mint are no more than two rows tall, so their queens are 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 #186

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3, 8 and 9 are the narrowest, touched by only 2 colors. Columns 8 and 9 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 5, meeting 5 colors, is the opposite case — the busiest line on #186 and the last one you should expect to crack.

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.

With ten queens to seat and ten regions competing over 100 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 10x10 boards quickly and players who stall halfway.

The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.

Why puzzle #186 is rated medium

The rating reflects how much the board concedes up front. Puzzle #186 has three regions confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 33 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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 #186 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 #186, start hunting for it near row 3.

Here is a shape check that gives nothing away: when #186 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.