10x10 Queens Puzzle #160

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 17-step reasoning is written out below.

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Step-by-step solution for puzzle #160

What follows is a complete solve of puzzle #160, 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 pairing argument on columns.

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

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

  1. Step 1. Red cannot leave row 6 — every one of its open squares (2 open squares) sits there. That row belongs to red, so every square on it owned by another colour is dead: 12 squares in all.

  2. Step 2. Sea green is down to 3 candidates in rows 9 and 10, columns 7 and 8, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about sea green itself.

  3. Step 3. Orange and sea green are now boxed into rows 9 and 10 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 10 squares at once.

    Puzzle #160 after step 3: 0 of 10 queens placed
    The board after step 3 — 0 of 10 queens placed.

  4. Step 4. Wherever khaki 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.

  5. Step 5. Slate and khaki 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 6 squares at once.

  6. Step 6. 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: indigo has one square left, so its queen takes row 1, column 8; sea green has one square left, so its queen takes row 10, column 7; mint has one square left, so its queen takes row 4, column 6; orange has one square left, so its queen takes row 9, column 2; column 9 is down to one square, putting a queen on row 8, column 9. 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 8, column 9.

    Puzzle #160 after step 6: 5 of 10 queens placed
    The board after step 6 — 5 of 10 queens placed.

  7. Step 7. Blue is down to 3 candidates in rows 2 and 3, columns 3–5, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about blue itself.

  8. Step 8. All of the amber region's remaining squares (3 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.

  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: pale yellow has one square left, so its queen takes row 7, column 5; red has one square left, so its queen takes row 6, column 3; blue has one square left, so its queen takes row 3, column 4; slate has one square left, so its queen takes row 5, column 10; amber has one square left, so its queen takes row 2, column 1. 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 2, column 1.

    Puzzle #160 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 #160

This is puzzle number 160 in the 10x10 collection, rated medium. The grid holds 100 squares split between ten colored regions, and a finished board carries ten queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.

There is exactly one way to finish #160, 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 #160 are: red (2 squares), sea green (3 squares), orange (5 squares), indigo (6 squares), slate (7 squares), khaki (7 squares), mint (12 squares), amber (16 squares), blue (18 squares), pale yellow (24 squares). An even split would hand every color 10 squares, so the red region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 24 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Red is working with 2 squares while pale yellow has 24 to spare, a gap of 22. Attack the cramped end and the roomy regions tend to resolve themselves.

Look at arrangement as well as area. Red is the densest region on this grid at 100% of a 1×2 frame; mint is the loosest, occupying only 60% of its 4×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

Region by region

Below is every color on #160 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

Red covers 2 squares, spanning 1 row and 2 columns (rows 6–6, columns 3–4). It never leaves row 6, so its queen is locked to that row.

Sea green occupies 3 squares within rows 9–10, columns 7–8.

Orange: 5 squares across rows 9–10, columns 1–4.

Indigo covers 6 squares, spanning 3 rows and 3 columns (rows 1–3, columns 8–10).

Slate occupies 7 squares within rows 3–7, columns 9–10.

Khaki: 7 squares across rows 7–10, columns 9–10.

Mint covers 12 squares, spanning 4 rows and 5 columns (rows 3–6, columns 5–9).

Amber occupies 16 squares within rows 1–6, columns 1–4.

Blue: 18 squares across rows 1–5, columns 3–8.

Pale yellow covers 24 squares, spanning 4 rows and 8 columns (rows 7–10, columns 1–8).

Where to start on this board

Red is the natural opening on #160: it is the smallest region on the grid at 2 squares, boxed into rows 6–6 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 red never leaves row 6, its queen must be in that row — and that instantly settles row 6 for everyone else. Sweep along it and X out every square that is not red. 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. Orange 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 #160

A quick way to find the next move is to count how many different colors touch each line. On this board, row 8 is the narrowest, touched by only 2 colors. Columns 1, 2, 5, 6 and 10 are the tightest columns, with 3 colors.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 9 rows. By contrast row 3 meets 5 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.

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.

Resist the plausible-looking queen. On #160 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.

Why puzzle #160 is rated medium

What earns a board its rating is the shape of its opening. Here that opening offers one single-line region, a 2-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 22. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

Stuck on #160 despite the medium 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.

Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 8 on this board.

As a sanity check rather than a spoiler: the finished solution to #160 puts 4 queens on the outer border and 6 queens in the interior, with none in a 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.

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 #160?

Puzzle #160 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 2 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #160 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #160, 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 #160?

Start with the red region — the most constrained color on this board, with 2 squares spanning rows 6–6 and columns 3–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.