10x10 Queens Puzzle #154

Difficulty: medium. 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 18-step reasoning is written out below.

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Step-by-step solution for puzzle #154

What follows is a complete solve of puzzle #154, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 10 queen placements and 8 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 pairing argument on columns.

Techniques this board needs: last square in a region, region shadow, pairing argument on columns, region confined to one line, pairing argument on rows, last square in a column.

Show the full walkthrough (14 steps) — contains the solution

  1. Step 1. Only row 1, column 10 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 10, the rest of the indigo region and every square adjacent to it — diagonals included.

  2. Step 2. The amber region is down to one open square, at row 2, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 3, the remaining amber squares, and the ring of neighbours around row 2, column 3.

  3. Step 3. Only row 4, column 9 is still open in mint, and since each colour must take a queen, that settles it. Everything else in row 4, in column 9, in mint and in the ring around that square is now dead.

  4. Step 4. Wherever blue ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.

  5. Step 5. Sea green is down to 5 candidates in rows 8–10, columns 6–8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about sea green itself.

    Puzzle #154 after step 5: 3 of 10 queens placed
    The board after step 5 — 3 of 10 queens placed.

  6. Step 6. Here is the pairing argument. The blue and pale yellow regions have no open squares outside columns 1 and 2 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 4 squares come off in one stroke.

  7. Step 7. Only row 8, column 4 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 4, in khaki and in the ring around that square is now dead.

  8. Step 8. All of the slate region's remaining squares (2 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.

  9. Step 9. Only row 3, column 1 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 1, the rest of the blue region and every square adjacent to it — diagonals included.

    Puzzle #154 after step 9: 5 of 10 queens placed
    The board after step 9 — 5 of 10 queens placed.

  10. Step 10. Wherever red 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.

  11. Step 11. Orange is down to 3 candidates in rows 7, 9, and 10, columns 5 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.

  12. Step 12. Here is the pairing argument. The red and pale yellow regions have no open squares outside rows 6 and 7 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 1 square come off in one stroke.

  13. Step 13. Wherever orange ends up among its 2 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

  14. Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: column 7 is down to one square, putting a queen on row 6, column 7; slate has one square left, so its queen takes row 5, column 5; pale yellow has one square left, so its queen takes row 7, column 2; orange has one square left, so its queen takes row 9, column 6; sea green has one square left, so its queen takes row 10, column 8. 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 10, column 8.

    Puzzle #154 after step 14: 10 of 10 queens placed
    The board after step 14 — 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 #154

Board #154 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.

As with every puzzle in this collection, #154 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.

How the colors divide this board

From smallest to largest, the regions on puzzle #154 are: indigo (1 square), amber (4 squares), slate (9 squares), pale yellow (9 squares), khaki (9 squares), orange (9 squares), mint (11 squares), sea green (11 squares), blue (13 squares), red (24 squares). An even split would hand every color 10 squares, so the indigo region is running 9 squares under that average while the red region is the roomiest area on the grid with 24 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 indigo, with 1 square, is under far more pressure than red with 24. A 23-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. Indigo is the tidiest shape on #154, packing 100% of its 1×1 bounding box, whereas mint straggles across its 5×4 box at just 55%. 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 medium board.

Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 10–10). It never leaves row 1, so its queen is locked to that row.

Amber occupies 4 squares within rows 1–2, columns 3–5.

Slate: 9 squares across rows 5–8, columns 3–6.

Pale yellow covers 9 squares, spanning 5 rows and 3 columns (rows 4–8, columns 1–3).

Khaki occupies 9 squares within rows 7–10, columns 1–4.

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

Mint covers 11 squares, spanning 5 rows and 4 columns (rows 1–5, columns 7–10).

Sea green occupies 11 squares within rows 8–10, columns 6–10.

Blue: 13 squares across rows 1–5, columns 1–4.

Red covers 24 squares, spanning 7 rows and 6 columns (rows 1–7, columns 5–10).

Where to start on this board

The most constrained color is where a solve should begin, and on #154 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 10 to column 10 — 1 row 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 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 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 #154

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3, 6, 9 and 10 are the narrowest, touched by only 3 colors. Columns 1, 2, 8 and 9 are the tightest columns, with 3 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 8 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

Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.

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.

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 #154 is rated medium

The medium label comes out of four measurable things about #154: two of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 23. Boards that score tight on those measures give up their first queen fast.

Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.

A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.

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 #154, start hunting for it near row 3.

Use this as a rough check on your progress: the completed board has 3 queens around the border and 7 queens in the middle, including 1 in a corner. 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.

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Questions about this puzzle

How hard is 10x10 Queens puzzle #154?

Puzzle #154 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #154 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #154, 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 #154?

Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.