10x10 Queens Puzzle #174

Difficulty: easy. 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 16-step reasoning is written out below.

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Step-by-step solution for puzzle #174

What follows is a complete solve of puzzle #174, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 10 queen placements and 6 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 confined to one line, region shadow, pairing argument on rows, pairing argument on columns.

Show the full walkthrough (9 steps) — contains the solution

  1. Step 1. Only row 2, column 7 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 7, the rest of the amber region and every square adjacent to it — diagonals included.

  2. Step 2. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.

  3. Step 3. The khaki region has 4 squares left, spread across rows 8–10, columns 1 and 2. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.

    Puzzle #174 after step 3: 1 of 10 queens placed
    The board after step 3 — 1 of 10 queens placed.

  4. Step 4. Wherever orange ends up among its 3 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.

  5. Step 5. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: sea green has one square left, so its queen takes row 10, column 3; orange has one square left, so its queen takes row 9, column 5; khaki has one square left, so its queen takes row 8, column 1; red has one square left, so its queen takes row 6, 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 6, column 2.

  6. Step 6. The pale yellow region has 2 squares left, spread across rows 5 and 7, columns 9 and 10. 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.

    Puzzle #174 after step 6: 5 of 10 queens placed
    The board after step 6 — 5 of 10 queens placed.

  7. Step 7. Here is the pairing argument. The slate and pale yellow regions have no open squares outside rows 5 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.

  8. Step 8. Indigo and pale yellow are now boxed into columns 9 and 10 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 3 squares at once.

  9. Step 9. 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: mint has one square left, so its queen takes row 4, column 8; 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 10; indigo has one square left, so its queen takes row 1, column 9; slate has one square left, so its queen takes row 5, column 6. 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 5, column 6.

    Puzzle #174 after step 9: 10 of 10 queens placed
    The board after step 9 — 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 #174

Board #174 of the 10x10 bank carries a easy 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, #174 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 #174 are: amber (1 square), orange (3 squares), sea green (3 squares), khaki (4 squares), indigo (7 squares), mint (7 squares), red (7 squares), pale yellow (11 squares), blue (25 squares), slate (32 squares). An even split would hand every color 10 squares, so the amber region is running 9 squares under that average while the slate region is the roomiest area on the grid with 32 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Amber has only 1 place to put its queen; slate has 32. The spread between them on this board is 31 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 amber region is the most compact color here, filling 100% of the 1×1 box that encloses it, while slate is the most scattered at 50% of its 8×8 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 #174, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

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

Orange covers 3 squares, spanning 2 rows and 2 columns (rows 9–10, columns 4–5).

Sea green occupies 3 squares within rows 9–10, columns 2–3.

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

Indigo covers 7 squares, spanning 2 rows and 4 columns (rows 1–2, columns 7–10).

Mint occupies 7 squares within rows 3–5, columns 8–10.

Red: 7 squares across rows 4–7, columns 1–3.

Pale yellow covers 11 squares, spanning 6 rows and 3 columns (rows 5–10, columns 8–10).

Blue occupies 25 squares within rows 1–5, columns 1–6.

Slate: 32 squares across rows 3–10, columns 2–9.

Where to start on this board

The most constrained color is where a solve should begin, and on #174 that is amber with 1 square. It reaches from row 2 to row 2 and from column 7 to column 7 — 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 amber never leaves row 2, its queen must be in that row — and that instantly settles row 2 for everyone else. Sweep along it and X out every square that is not amber. 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, sea green and orange 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 #174

A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Column 6 is the tightest column, with 2 colors.

Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 10 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

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.

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.

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 #174 is rated easy

The rating reflects how much the board concedes up front. Puzzle #174 has two regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 31 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

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 #174 the place to look for one is around row 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.

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Questions about this puzzle

How hard is 10x10 Queens puzzle #174?

Puzzle #174 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #174 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #174, 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 #174?

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