8x8 Queens Puzzle #172
Difficulty: easy. Place 8 queens on this 8x8 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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #172
What follows is a complete solve of puzzle #172, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 8 queen placements and 11 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, line confined to one region, pairing argument on rows, pairing argument on columns, last square in a region.
Show the full walkthrough (12 steps) — contains the solution
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: 8 squares in all.
Step 2. Mint is down to 5 candidates in rows 3 and 4, columns 4–7, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about mint itself.
Step 3. The red region has 3 squares left, spread across rows 6 and 7, columns 7 and 8. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 4. All of the khaki region's remaining squares (3 open squares) lie in row 8. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.
The board after step 4 — 0 of 8 queens placed. Step 5. Pale yellow is down to 3 candidates in rows 6 and 7, columns 4–6, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 6. All of the red region's remaining squares (2 open squares) lie in column 8. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
Step 7. Wherever blue ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 8. Mint is down to 3 candidates in rows 3 and 4, columns 4 and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.
The board after step 8 — 0 of 8 queens placed. Step 9. Row 2 has only amber squares left (4 open squares). The row needs a queen and only amber can provide one, which pins that colour to this row and kills its squares elsewhere: 4 squares.
Step 10. 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: 6 squares come off in one stroke.
Step 11. Blue, mint, pale yellow, and khaki are now boxed into columns 4–7 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 5 squares at once.
Step 12. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 3; indigo has one square left, so its queen takes row 1, column 1; slate has one square left, so its queen takes row 5, column 2; blue has one square left, so its queen takes row 3, column 7; mint has one square left, so its queen takes row 4, column 5; khaki has one square left, so its queen takes row 8, column 6; pale yellow has one square left, so its queen takes row 7, column 4; red has one square left, so its queen takes row 6, 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 6, column 8.
The board after step 12 — 8 of 8 queens placed.
That completes the board: 8 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 8x8 Queens puzzle #172
This is puzzle number 172 in the 8x8 collection, rated easy. The grid holds 64 squares split between eight colored regions, and a finished board carries eight 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 #172, 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 eight regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #172 are: indigo (2 squares), red (3 squares), khaki (4 squares), mint (5 squares), pale yellow (8 squares), blue (11 squares), amber (15 squares), slate (16 squares). An even split would hand every color 8 squares, so the indigo region is running 6 squares under that average while the slate region is the roomiest area on the grid with 16 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 2 squares while slate has 16 to spare, a gap of 14. Attack the cramped end and the roomy regions tend to resolve themselves.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #172, packing 100% of its 1×2 bounding box, whereas pale yellow straggles across its 3×6 box at just 44%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Here is the full footprint of every color on puzzle #172, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Indigo covers 2 squares, spanning 1 row and 2 columns (rows 1–1, columns 1–2). It never leaves row 1, so its queen is locked to that row.
Red occupies 3 squares within rows 6–7, columns 7–8.
Khaki: 4 squares across rows 8–8, columns 5–8. It never leaves row 8, so its queen is locked to that row.
Mint covers 5 squares, spanning 2 rows and 4 columns (rows 3–4, columns 4–7).
Pale yellow occupies 8 squares within rows 6–8, columns 1–6.
Blue: 11 squares across rows 1–6, columns 6–8.
Amber covers 15 squares, spanning 4 rows and 7 columns (rows 1–4, columns 1–7).
Slate occupies 16 squares within rows 4–7, columns 1–5.
Where to start on this board
The most constrained color is where a solve should begin, and on #172 that is indigo with 2 squares. It reaches from row 1 to row 1 and from column 1 to column 2 — 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. Khaki, mint and red 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 #172
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 5 and 8 are the narrowest, touched by only 2 colors. Columns 3 and 8 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 7 rows. By contrast row 6 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
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.
With eight queens to seat and eight regions competing over 64 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 8x8 boards quickly and players who stall halfway.
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 #172 is rated easy
The rating reflects how much the board concedes up front. Puzzle #172 has two regions confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 14 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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 easy board from the rest.
Stuck on #172 despite the easy 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 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.
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 2 on this board.
As a sanity check rather than a spoiler: the finished solution to #172 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.
How to play Queens
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- 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.
- 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 8x8 Queens puzzle #172?
Puzzle #172 is rated easy. Its smallest color region holds 2 squares, 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 8x8 puzzle #172 have only one solution?
Yes. Every puzzle in the 8x8 collection, including #172, has been verified to have exactly one valid arrangement of eight queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #172?
Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–1 and columns 1–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.