10x10 Queens Puzzle #117

About 10x10 Queens puzzle #117

Puzzle #117 is one of the easy-rated boards in the 10x10 set. Ten colors divide its 100 squares, and you win by placing ten queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.

There is exactly one way to finish #117, 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 #117 are: indigo (1 square), red (1 square), sea green (1 square), amber (4 squares), mint (7 squares), orange (10 squares), khaki (11 squares), slate (14 squares), blue (17 squares), pale yellow (34 squares). An even split would hand every color 10 squares, so the indigo region is running 9 squares under that average while the pale yellow region is the roomiest area on the grid with 34 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 1 place to put its queen; pale yellow has 34. 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.

Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×1 box that encloses it, while blue is the most scattered at 49% of its 7×5 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 #117, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Indigo: 1 square across rows 1–1, columns 10–10. It never leaves row 1, so its queen is locked to that row.

Red covers 1 square, spanning 1 row and 1 column (rows 6–6, columns 3–3). It never leaves row 6, so its queen is locked to that row.

Sea green occupies 1 square within rows 10–10, columns 8–8. It never leaves row 10, so its queen is locked to that row.

Amber: 4 squares across rows 1–3, columns 1–2.

Mint covers 7 squares, spanning 4 rows and 2 columns (rows 1–4, columns 6–7).

Orange occupies 10 squares within rows 8–10, columns 3–8.

Khaki: 11 squares across rows 7–10, columns 1–4.

Slate covers 14 squares, spanning 7 rows and 4 columns (rows 2–8, columns 3–6).

Blue occupies 17 squares within rows 1–7, columns 1–5.

Pale yellow: 34 squares across rows 1–10, columns 5–10.

Where to start on this board

Begin where the choices are fewest. On this board that means indigo, which owns just 1 square inside a 1×1 window running from row 1 to row 1 and column 10 to column 10. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Red and sea green 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 #117

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

The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 3, meeting 5 colors, is the opposite case — the busiest line on #117 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.

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.

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 #117 is rated easy

The rating reflects how much the board concedes up front. Puzzle #117 has six regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 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.

The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a easy board from the rest.

Stuck on #117 despite the easy 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 #117 the place to look for one is around column 9.

Here is a shape check that gives nothing away: when #117 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.