7x7 Queens Puzzle #196
Difficulty: hard. Place 7 queens on this 7x7 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 12-step reasoning is written out below.
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Step-by-step solution for puzzle #196
What follows is a complete solve of puzzle #196, worked the way a person works it rather than the way a computer does. It runs to 12 deductions — 7 queen placements and 5 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 shadow, region confined to one line, last square in a region.
Show the full walkthrough (9 steps) — contains the solution
Step 1. Wherever indigo 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 2. All of the amber region's remaining squares (2 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 6 squares come off.
Step 3. Blue cannot leave row 3 — every one of its open squares (2 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 7 squares in all.
The board after step 3 — 0 of 7 queens placed. Step 4. 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: 4 squares come off.
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: amber has one square left, so its queen takes row 2, column 7; blue has one square left, so its queen takes row 3, column 5; indigo has one square left, so its queen takes row 1, column 4; mint has one square left, so its queen takes row 4, 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 4, column 1.
Step 6. The slate region has 3 squares left, spread across rows 5 and 6, columns 2 and 3. 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.
The board after step 6 — 4 of 7 queens placed. Step 7. The pale yellow region is down to one open square, at row 7, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 2, the remaining pale yellow squares, and the ring of neighbours around row 7, column 2.
Step 8. Only row 5, column 3 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 3, in slate and in the ring around that square is now dead.
Step 9. The red region is down to one open square, at row 6, column 6. Every colour holds exactly one queen, so that square is forced. Row 6 and column 6 are now spoken for, along with the squares touching that queen.
The board after step 9 — 7 of 7 queens placed.
That completes the board: 7 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 7x7 Queens puzzle #196
This is puzzle number 196 in the 7x7 collection, rated hard. The grid holds 49 squares split between seven colored regions, and a finished board carries seven 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 #196, 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 seven regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #196 are: amber (2 squares), indigo (3 squares), mint (4 squares), pale yellow (5 squares), blue (6 squares), red (11 squares), slate (18 squares). An even split would hand every color 7 squares, so the amber region is running 5 squares under that average while the slate region is the roomiest area on the grid with 18 squares.
Every color takes one queen and one only, whatever its size — which is why the 16 square difference between amber and slate is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
A region's outline can constrain it more than its size does. Here amber is the most solid, filling 100% of its 2×1 box, and blue the most diffuse at 50% of its 4×3 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
Amber covers 2 squares, spanning 2 rows and 1 column (rows 1–2, columns 7–7). It sits wholly inside column 7, so its queen must be in that column.
Indigo occupies 3 squares within rows 1–2, columns 4–5.
Mint: 4 squares across rows 1–4, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Pale yellow covers 5 squares, spanning 3 rows and 3 columns (rows 5–7, columns 1–3).
Blue occupies 6 squares within rows 1–4, columns 5–7.
Red: 11 squares across rows 4–7, columns 4–7.
Slate covers 18 squares, spanning 7 rows and 4 columns (rows 1–7, columns 2–5).
Where to start on this board
Open on the tightest color. Here that is amber, confined to 2 squares between rows 1 and 2 and columns 7 and 7. Its queen has nowhere else to go, which means those 2 rows and 1 column carry more constraint than anywhere else on #196.
Because amber stays inside column 7, its queen belongs to that column and no other region may use it. Cross out the rest of column 7 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
Work outward to the other flat regions next. Indigo 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 #196
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3, 5, 6 and 7 are the narrowest, touched by only 3 colors. Columns 1, 2, 3 and 6 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 6 rows. Row 2 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.
The whole middle game reduces to two sweeps. Sweep the seven colors looking for one with a single square left; sweep the seven rows and seven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 7x7 solve from a stalled one.
A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.
Why puzzle #196 is rated hard
Ratings track how generous the opening is. On #196 you get two regions pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 3 colors, and a 16-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.
Stuck on #196 despite the hard 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 7x7
At 7x7 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.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #196 is column 1.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 3 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
- 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 7x7 Queens puzzle #196?
Puzzle #196 is rated hard. Its smallest color region holds 2 squares, 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 7x7 puzzle #196 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #196, has been verified to have exactly one valid arrangement of seven queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #196?
Start with the amber region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.