8x8 Queens Puzzle #114

About 8x8 Queens puzzle #114

Board #114 of the 8x8 bank carries a medium rating. Its 64 squares are shared between eight colored regions, and the solution places eight queens: one per row, one per column, one per region, none of them side by side or corner to corner.

Puzzle #114 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.

How the colors divide this board

From smallest to largest, the regions on puzzle #114 are: khaki (2 squares), indigo (3 squares), pale yellow (3 squares), amber (9 squares), mint (9 squares), red (9 squares), slate (11 squares), blue (18 squares). An even split would hand every color 8 squares, so the khaki region is running 6 squares under that average while the blue region is the roomiest area on the grid with 18 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Khaki has only 2 places to put its queen; blue has 18. The spread between them on this board is 16 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×3 box, and mint the most diffuse at 50% of its 6×3 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

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 medium board.

Khaki: 2 squares across rows 8–8, columns 3–4. It never leaves row 8, so its queen is locked to that row.

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

Pale yellow occupies 3 squares within rows 7–8, columns 5–6.

Amber: 9 squares across rows 1–3, columns 5–8.

Mint covers 9 squares, spanning 6 rows and 3 columns (rows 2–7, columns 5–7).

Red occupies 9 squares within rows 6–8, columns 1–4.

Slate: 11 squares across rows 3–8, columns 6–8.

Blue covers 18 squares, spanning 6 rows and 4 columns (rows 1–6, columns 1–4).

Where to start on this board

Khaki is the natural opening on #114: it is the smallest region on the grid at 2 squares, boxed into rows 8–8 and columns 3–4. Since one of those squares must hold a queen, that block of 1 row and 2 columns is the most contested part of the board.

Because khaki never leaves row 8, its queen must be in that row — and that instantly settles row 8 for everyone else. Sweep along it and X out every square that is not khaki. 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. Indigo and pale yellow 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 #114

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

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 eight colors looking for one with a single square left; sweep the eight rows and eight columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 8x8 solve from a stalled one.

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 #114 is rated medium

The medium label comes out of four measurable things about #114: two of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 16. Boards that score tight on those measures give up their first queen fast.

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 medium board from the rest.

When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.

Techniques that work well at 8x8

At 8x8 the whole grid stays within a single glance, which makes systematic scanning practical. Read the rows in order and count the colors in each — a row touched by two colors is far more constrained than one touched by five, and that is usually where the next forced move hides. The board is small enough that re-scanning from the top after each placement costs almost nothing.

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 #114 the place to look for one is around column 8.

As a sanity check rather than a spoiler: the finished solution to #114 puts 3 queens on the outer border and 5 queens in the interior, 1 of them in an actual 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.