10x10 Queens Puzzle #78

Difficulty: hard. Place 10 queens on this 10x10 board so that no two share a row, column or color region and none touch — including diagonally. This puzzle has exactly one solution, and it has been verified solvable by deduction alone: the complete 17-step reasoning is written out below.

Previous puzzle · All 10x10 puzzles · Next puzzle

Step-by-step solution for puzzle #78

What follows is a complete solve of puzzle #78, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 10 queen placements and 7 elimination steps that narrow the board between them. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the region shadow.

Techniques this board needs: region confined to one line, last square in a region, region shadow.

Show the full walkthrough (11 steps) — contains the solution

  1. Step 1. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 10 squares in all.

  2. Step 2. The amber region is down to one open square, at row 2, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 10, the remaining amber squares, and the ring of neighbours around row 2, column 10.

  3. Step 3. Blue cannot leave row 3 — every one of its open squares (5 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 3 squares in all.

  4. Step 4. The mint region is down to one open square, at row 4, column 1. Every colour holds exactly one queen, so that square is forced. Row 4 and column 1 are now spoken for, along with the squares touching that queen.

    Puzzle #78 after step 4: 2 of 10 queens placed
    The board after step 4 — 2 of 10 queens placed.

  5. Step 5. Only row 5, column 3 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 3, the rest of the slate region and every square adjacent to it — diagonals included.

  6. Step 6. The khaki region has 3 squares left, spread across rows 8 and 9, columns 7 and 8. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.

  7. Step 7. Wherever pale yellow ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

    Puzzle #78 after step 7: 3 of 10 queens placed
    The board after step 7 — 3 of 10 queens placed.

  8. Step 8. Sea green is down to 3 candidates in rows 9 and 10, columns 8 and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about sea green itself.

  9. Step 9. Khaki cannot leave row 8 — every one of its open squares (2 open squares) sits there. That row belongs to khaki, so every square on it owned by another colour is dead: 5 squares in all.

  10. Step 10. All of the pale yellow region's remaining squares (2 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.

  11. Step 11. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: sea green has one square left, so its queen takes row 10, column 8; khaki has one square left, so its queen takes row 8, column 7; indigo has one square left, so its queen takes row 1, column 6; red has one square left, so its queen takes row 6, column 5; blue has one square left, so its queen takes row 3, column 4; pale yellow has one square left, so its queen takes row 7, column 9; orange has one square left, so its queen takes row 9, column 2. After each one, clear that row, that column, the rest of the colour and the ring of neighbours before looking for the next. The last of them seats the queen on row 9, column 2.

    Puzzle #78 after step 11: 10 of 10 queens placed
    The board after step 11 — 10 of 10 queens placed.

That completes the board: 10 queens seated, one in every row, one in every column and one in every colour, with no two of them touching. Notice that nothing in the sequence above required a guess — each placement followed from an elimination already on the grid, which is what “single solution, solvable by logic” actually means in practice.

About 10x10 Queens puzzle #78

Board #78 of the 10x10 bank carries a hard 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, #78 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 #78 are: indigo (2 squares), khaki (3 squares), sea green (3 squares), amber (4 squares), slate (4 squares), mint (8 squares), blue (14 squares), pale yellow (17 squares), red (21 squares), orange (24 squares). An even split would hand every color 10 squares, so the indigo region is running 8 squares under that average while the orange region is the roomiest area on the grid with 24 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 2 places to put its queen; orange has 24. The spread between them on this board is 22 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 1×2 box that encloses it, while pale yellow is the most scattered at 47% of its 9×4 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

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: 2 squares across rows 1–1, columns 6–7. It never leaves row 1, so its queen is locked to that row.

Khaki covers 3 squares, spanning 2 rows and 2 columns (rows 8–9, columns 7–8).

Sea green occupies 3 squares within rows 9–10, columns 8–9.

Amber: 4 squares across rows 1–2, columns 8–10.

Slate covers 4 squares, spanning 2 rows and 3 columns (rows 4–5, columns 2–4).

Mint occupies 8 squares within rows 1–4, columns 1–3.

Blue: 14 squares across rows 1–3, columns 3–8.

Pale yellow covers 17 squares, spanning 9 rows and 4 columns (rows 2–10, columns 7–10).

Red occupies 21 squares within rows 4–10, columns 4–8.

Orange: 24 squares across rows 5–10, columns 1–5.

Where to start on this board

The most constrained color is where a solve should begin, and on #78 that is indigo with 2 squares. It reaches from row 1 to row 1 and from column 6 to column 7 — 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 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. Amber, slate, khaki 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 #78

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

Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 9 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.

Working through the middle game

The middle of #78 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.

Keep alternating between two views of the board. In the region view you ask which of the ten colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 10x10 board unwinds.

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 #78 is rated hard

Ratings track how generous the opening is. On #78 you get one region pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 3 colors, and a 22-square gap between the largest and smallest regions. Those four numbers largely explain the hard 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 #78 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 #78 the place to look for one is around column 1.

Use this as a rough check on your progress: the completed board has 4 queens around the border and 6 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.

How to play Queens

  1. The board is an N×N grid divided into N colored regions.
  2. Place exactly one queen (♛) in each row, each column, and each colored region.
  3. No two queens may touch each other — not horizontally, vertically, or diagonally.
  4. Tap once to mark a square with an X (where a queen can't go), tap again to place a queen, and once more to clear it.
  5. There is always exactly one valid solution. Use logic, never guessing, to find it.

Full rules · Strategy library · Puzzle statistics

Questions about this puzzle

How hard is 10x10 Queens puzzle #78?

Puzzle #78 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #78 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #78, has been verified to have exactly one valid arrangement of ten queens, and it can always be reached by logic alone — no guessing or backtracking needed.

Where should I place my first queen on puzzle #78?

Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–1 and columns 6–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.